OFFSET
0,3
REFERENCES
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
Nicholas Dub, Enumeration of triangulations modulo symmetries and of rooted triangulations counted by their number of (d - 2)-simplices in dimension d ≥ 2, tel-03641958 [cs.OH], Université Paris-Nord - Paris XIII, 2021.
Sean A. Irvine, Illustration of initial terms
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)
FORMULA
G.f.: exp(Sum_{k >= 1} F(x^k) / k) where F(x) is the g.f. for A002831. - Sean A. Irvine, Sep 09 2014
MATHEMATICA
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t k; s += t]; s!/m];
b[k_, q_] := If[OddQ[q], If[OddQ[k], 0, j = k/2; q^j (2 j)!/(j! 2^j)], Sum[ Binomial[k, 2 j] q^j (2 j)!/(j! 2^j), {j, 0, Quotient[k, 2]}]];
pm[v_] := Module[{p = Total[x^v]}, Product[b[Coefficient[p, x, i], i], {i, 1, Exponent[p, x]}]];
a[n_] := Module[{s = 0}, Do[s += permcount[p] pm[p]^3, {p, IntegerPartitions[2 n]}]; s/(2 n)!];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 30}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)
PROG
(PARI)
b(k, r) = {if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
g(n, k)={sum(r=0, n\k, x^(k*r)*b(k, r)^3/(r!*k^r)) + O(x*x^n)}
seq(n)={Vec(substpol(prod(k=1, 2*n, g(2*n, k)), x^2, x))} \\ Andrew Howroyd, Dec 14 2017; updated May 02 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(7)-a(8) from Sean A. Irvine, Sep 08 2014
Terms a(9) and beyond from Andrew Howroyd, Dec 14 2017
a(0)=1 prepended by Andrew Howroyd, May 02 2023
STATUS
approved