OFFSET
0,9
COMMENTS
a(n) = A116376(3*n). - Reinhard Zumkeller, Feb 15 2006
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
James Mc Laughlin, Andrew V. Sills, Peter Zimmer, Rogers-Ramanujan-Slater Type Identities , arXiv:1901.00946 [math.NT]
FORMULA
a(n) = 1/n*Sum_{k=1..n} A078182(k)*a(n-k), a(0) = 1. - Vladeta Jovovic, Nov 21 2002
Euler transform of period 3 sequence [ 0, 1, 0, ...]. - Michael Somos, Jul 24 2007
a(n) ~ Gamma(2/3) * exp(sqrt(2*n)*Pi/3) / (2^(11/6) * sqrt(3) * Pi^(1/3) * n^(5/6)) * (1 + (Pi/72 - 5/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
G.f.: A(x) = Sum_{n >= 0} x^(n*(3*n-1))/Product_{k = 1..n} ((1 - x^(3*k)) *(1 - x^(3*k-1))). (Set z = x^2 and q = x^3 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
MAPLE
g:=add(x^(n*(3*n-1))/mul((1-x^(3*k))*(1-x^(3*k-1)), k = 1..n), n = 0..6): gser:=series(g, x, 101): seq(coeff(gser, x, n), n = 0..100); # Peter Bala, Feb 02 2021
MATHEMATICA
nmax=100; CoefficientList[Series[Product[1/(1-x^(3*k+2)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 3] == 2, Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly2 = Take[poly, {2, nmax + 1}]; poly3 = 1 + Sum[poly2[[n]]*x^n, {n, 1, Length[poly2]}]; CoefficientList[Series[1/poly3, {x, 0, Length[poly2]}], x] (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 50; s = Range[0, nmax/3]*3 + 2;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)
PROG
(PARI) {a(n)= if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - (k%3==2) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jul 24 2007 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved