The On-Line Encyclopedia of Integer Sequences (OEIS)
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1 The On-Line Encyclopedia of Integer Sequences (OEIS) Thotsaporn Aek Thanatipanonda Mahidol University, International College September 26, 2018 of Integer Sequences (OEIS) September 26, / 14
2 Introduction OEIS is the database of integer sequences. For example, if I count some combinatorial object and get the first few terms to be 1, 3, 13, 63,.... I can find out more information from this website. of Integer Sequences (OEIS) September 26, / 14
3 Introduction OEIS is the database of integer sequences. For example, if I count some combinatorial object and get the first few terms to be 1, 3, 13, 63,.... I can find out more information from this website. We found out that the number of King walk from (0, 0) to (n, n) grid is The recurrence for a n is a n := n k=0 ( ) n k ( ) n + k (n + 2)a n (6n + 9)a n 1 + (n + 1)a n 2 = 0 k hotsaporn Aek Thanatipanonda (MahidolThe University, On-LineInternational EncyclopediaCollege) of Integer Sequences (OEIS) September 26, / 14
4 Introduction Neil Sloane, founder of OEIS of Integer Sequences (OEIS) September 26, / 14
5 Time line Neil Sloane was born on Oct. 10, : start to collect integer sequences 1969: work at AT&T Bell labs (Now Nokia) 1973: publish A handbook of integer sequences which contains 2372 sequences. 1995: book version of The encyclopedia of integer sequences (5488 sequences) 1996: Internet version of The encyclopedia of integer sequences 2004: OEIS hits 100,000 sequences of Integer Sequences (OEIS) September 26, / 14
6 Fun Fact There are about 315,271 sequences. The number of comments keeps increasing, and at present averages between 30 and 60 a day. Web traffic on all my web pages averages about 600,000 page-downloads per month. hotsaporn Aek Thanatipanonda (MahidolThe University, On-LineInternational EncyclopediaCollege) of Integer Sequences (OEIS) September 26, / 14
7 Fun Fact There are about 315,271 sequences. The number of comments keeps increasing, and at present averages between 30 and 60 a day. Web traffic on all my web pages averages about 600,000 page-downloads per month. Anyone can submit the sequence to OEIS. It is not as hard as you might think. In fact, you can submit the sequence automatically as well. hotsaporn Aek Thanatipanonda (MahidolThe University, On-LineInternational EncyclopediaCollege) of Integer Sequences (OEIS) September 26, / 14
8 Fun Fact There are about 315,271 sequences. The number of comments keeps increasing, and at present averages between 30 and 60 a day. Web traffic on all my web pages averages about 600,000 page-downloads per month. Anyone can submit the sequence to OEIS. It is not as hard as you might think. In fact, you can submit the sequence automatically as well. Welcome page in Thai language! hotsaporn Aek Thanatipanonda (MahidolThe University, On-LineInternational EncyclopediaCollege) of Integer Sequences (OEIS) September 26, / 14
9 Fun Fact There are about 315,271 sequences. The number of comments keeps increasing, and at present averages between 30 and 60 a day. Web traffic on all my web pages averages about 600,000 page-downloads per month. Anyone can submit the sequence to OEIS. It is not as hard as you might think. In fact, you can submit the sequence automatically as well. Welcome page in Thai language! Music generated by a sequence! hotsaporn Aek Thanatipanonda (MahidolThe University, On-LineInternational EncyclopediaCollege) of Integer Sequences (OEIS) September 26, / 14
10 Fun Fact There are about 315,271 sequences. The number of comments keeps increasing, and at present averages between 30 and 60 a day. Web traffic on all my web pages averages about 600,000 page-downloads per month. Anyone can submit the sequence to OEIS. It is not as hard as you might think. In fact, you can submit the sequence automatically as well. Welcome page in Thai language! Music generated by a sequence! 2D sequences or fraction sequences are in OEIS too. Just type them in! hotsaporn Aek Thanatipanonda (MahidolThe University, On-LineInternational EncyclopediaCollege) of Integer Sequences (OEIS) September 26, / 14
11 Some famous/not so famous sequences A1: 0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5,... of Integer Sequences (OEIS) September 26, / 14
12 Some famous/not so famous sequences A1: 0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5,... (Number of groups of order n) A43: 2, 3, 5, 7, 13, 17, 19,... of Integer Sequences (OEIS) September 26, / 14
13 Some famous/not so famous sequences A1: 0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5,... (Number of groups of order n) A43: 2, 3, 5, 7, 13, 17, 19,... (Mersenne primes) A326: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176,... of Integer Sequences (OEIS) September 26, / 14
14 Some famous/not so famous sequences A1: 0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5,... (Number of groups of order n) A43: 2, 3, 5, 7, 13, 17, 19,... (Mersenne primes) A326: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176,... (Pentagonal numbers: a(n) = n(3n 1) 2 ) A787: 0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609,... of Integer Sequences (OEIS) September 26, / 14
15 Neil Sloane s First Love The sequence database was begun by Neil J. A. Sloane in early 1964 when he was a graduate student at Cornell University in Ithaca, NY. He had encountered a sequence of numbers while working on his dissertation, namely 1, 8, 78, 944,... (now entry A in the OEIS), and was looking for a formula for the n-th term, in order to determine the rate of growth of the terms. of Integer Sequences (OEIS) September 26, / 14
16 Neil Sloane s First Love This sequence now has been expanded to more than quarter-million sequences and is expressible by the formula n 2 n k (n 1)! k!. Sloane and John Riordan (1969) showed that this is the sum of the total heights, taken over all labeled rooted tree with n vertices, divided by n. k=0 of Integer Sequences (OEIS) September 26, / 14
17 Labeled Rooted Trees Here we will follow Doron Zeilberger treatment, Going Back to Neil Sloane s FIRST LOVE (OEIS Sequence A435), to this formula by Sloane. Arthur Cayley famously proved that the number of labeled trees on n vertices is n n 2, hence the number of labeled rooted tree is n n n 2 = n n 1. We will first prove this result (the method due to Andre Joyal) and expand the method to the result of Sloane and more. of Integer Sequences (OEIS) September 26, / 14
18 Proof of Cayley s Formula Theorem (Borchardt, 1860) Let r(n) be the number of labeled rooted trees with n vertices. We have r(n) = n n 1. of Integer Sequences (OEIS)September 26, / 14
19 Proof of Cayley s Formula Theorem (Borchardt, 1860) Let r(n) be the number of labeled rooted trees with n vertices. We have r(n) = n n 1. of Integer Sequences (OEIS)September 26, / 14
20 Proof of Cayley s Formula Outline of the Proof 1 Let R(x) := r(n) n=0 n! x n. Claim the functional equation: R(x) k R(x) = x k! k=0 = xe R(x). of Integer Sequences (OEIS)September 26, / 14
21 Proof of Cayley s Formula Outline of the Proof 1 Let R(x) := r(n) n=0 n! x n. Claim the functional equation: R(x) k R(x) = x k! k=0 = xe R(x). 2 Apply Lagrange Inversion Theorem: If R(x) and Φ(z) are formal power series which starting at x and z 0 respectively, then R(x) = xφ(r(x)) implies [x n ]R(x) = 1 n [zn 1 ]Φ(z) n. of Integer Sequences (OEIS)September 26, / 14
22 Proof of Sloane s Formula Theorem Let s(n) be the sum of the total heights, taken over all labeled rooted tree with n vertices. We have n 2 n k s(n) = n! k!. k=0 of Integer Sequences (OEIS)September 26, / 14
23 Proof of Sloane s Formula Outline of the Proof 1 Consider J n (y) = T y TotalHeight(T ). Note J n (1) = r(n), number of rooted labeled tree. Then defined the formal power series J(x, y) = Claimed the functional equation n=1 J n (y) x n n!. J(x, y) = xe J(xy,y). of Integer Sequences (OEIS)September 26, / 14
24 Proof of Sloane s Formula 2 It is too much too ask for formula for J(x, y) explicitly. But for Sloane s formula, we only need J y (x, 1). This can be done by the chain rule, which gives us: J y (x, 1) = R(x)2 [1 R(x)] 2. 3 Apply General Lagrange Inversion Theorem: If R(x) and Φ(z) are formal power series which starting at x and z 0 respectively, and G(z) is yet another formal power series, then R(x) = xφ(r(x)) implies [x n ]G(R(x)) = 1 n [zn 1 ]G (z)φ(z) n. of Integer Sequences (OEIS)September 26, / 14
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