cp's OEIS Frontend

A294500 Binomial transform of the number of planar partitions (A000219).

%I A294500 #15 Nov 02 2023 06:58:48
%S A294500 1,2,6,19,60,185,559,1662,4875,14134,40564,115370,325465,911355,
%T A294500 2534595,7004827,19246626,52596377,143006632,386984573,1042537831,
%U A294500 2796803110,7473161196,19893461042,52767059608,139488323734,367540167625,965445514862,2528516552660
%N A294500 Binomial transform of the number of planar partitions (A000219).
%C A294500 Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
%C A294500 Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
%C A294500 g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
%C A294500 Special cases:
%C A294500 p < 1/2, g(n) = 0
%C A294500 p = 1/2, g(n) = r^2/16
%C A294500 p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
%C A294500 p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536
%C A294500 p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
%C A294500 p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625
%H A294500 Vaclav Kotesovec, <a href="/A294500/b294500.txt">Table of n, a(n) for n = 0..2930</a>
%F A294500 a(n) = Sum_{k=0..n} binomial(n,k) * A000219(k).
%F A294500 a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) - Zeta(3)/12) * 2^(n + 7/18) * Zeta(3)^(7/36) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.
%F A294500 G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)). - _Ilya Gutkovskiy_, Aug 20 2018
%t A294500 nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]
%Y A294500 Cf. A218481, A266232, A294501, A294502, A294504.
%K A294500 nonn
%O A294500 0,2
%A A294500 _Vaclav Kotesovec_, Nov 01 2017