A339987 The number of labeled graphs on 2n vertices that share the same degree sequence as any unrooted binary tree on 2n vertices.
1, 4, 90, 8400, 1426950, 366153480, 134292027870, 67095690261600, 43893900947947050, 36441011093916429000, 37446160423265535041100, 46669357647008722700474400, 69367722399061403579194432500, 121238024532751529573125745790000, 246171692450596203263023527657431250
Offset: 1
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..100 (terms 1..40 from Atabey Kaygun)
- Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence.
- Atabey Kaygun, Common LISP program that generates the sequence.
- M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.
Programs
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PARI
\\ See Links in A295193 for GraphsByDegreeSeq. a(n) = {if(n==1, 1, my(d=2*n-4, M=GraphsByDegreeSeq(n-1, 3, (p,r)-> subst(deriv(p),x,1) >= d-6*r), z=(2*n)!/(n-1)!); sum(i=1, matsize(M)[1], my(p=M[i,1], r=(subst(deriv(p), x, 1)-d)/2); M[i,2]*z / (2^polcoef(p,1) * 6^polcoef(p,0) * 2^r * r!)))} \\ Andrew Howroyd, Mar 01 2023
Formula
Conjectured recurrence: 32*(1 + n)*(2 + n)*(1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11589 + 10844*n + 3300*n^2 + 328*n^3)*a(n) - 8*(2 + n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(148119 + 232328*n + 129460*n^2 + 30664*n^3 + 2624*n^4)*a(n+1) - 16*(3 + n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(341634 + 712135*n + 569267*n^2 + 219308*n^3 + 40852*n^4 + 2952*n^5)*a(n+2) + 8*(4 + n)*(7 + 2*n)*(9 + 2*n)*(527520 + 1057879*n + 818282*n^2 + 306380*n^3 + 55672*n^4 + 3936*n^5)*a(n+3) - 2*(5 + n)*(9 + 2*n)*(601452 + 1117119*n + 786236*n^2 + 264028*n^3 + 42472*n^4 + 2624*n^5)*a(n+4) + 3*(4 + n)*(6 + n)*(3717 + 5228*n + 2316*n^2 + 328*n^3)*a(n+5) = 0. - Manuel Kauers and Christoph Koutschan, Mar 01 2023
Conjecture: a(n) ~ 2^(5*n - 1/2) * n^(2*n - 3/2) / (sqrt(Pi) * 3^(n-1) * exp(2*n + 21/16)), based on the recurrence by Manuel Kauers and Christoph Koutschan. - Vaclav Kotesovec, Mar 07 2023
Comments