%I #15 Nov 02 2023 06:58:48

%S 1,2,6,19,60,185,559,1662,4875,14134,40564,115370,325465,911355,

%T 2534595,7004827,19246626,52596377,143006632,386984573,1042537831,

%U 2796803110,7473161196,19893461042,52767059608,139488323734,367540167625,965445514862,2528516552660

%N Binomial transform of the number of planar partitions (A000219).

%C Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then

%C Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where

%C g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).

%C Special cases:

%C p < 1/2, g(n) = 0

%C p = 1/2, g(n) = r^2/16

%C p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81

%C 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 p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))

%C 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 Vaclav Kotesovec, <a href="/A294500/b294500.txt">Table of n, a(n) for n = 0..2930</a>

%F a(n) = Sum_{k=0..n} binomial(n,k) * A000219(k).

%F 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 G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)). - _Ilya Gutkovskiy_, Aug 20 2018

%t 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 Cf. A218481, A266232, A294501, A294502, A294504.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 01 2017