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

1, 2, 6, 19, 60, 185, 559, 1662, 4875, 14134, 40564, 115370, 325465, 911355, 2534595, 7004827, 19246626, 52596377, 143006632, 386984573, 1042537831, 2796803110, 7473161196, 19893461042, 52767059608, 139488323734, 367540167625, 965445514862, 2528516552660

Offset

0,2

Comments

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

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

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

Special cases:

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

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

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

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

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

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2930

Formula

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

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.

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

Mathematica

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}]

Crossrefs

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

Sequence in context: A014346 A183188 A118364 * A208481 A052544 A204200

Adjacent sequences: A294497 A294498 A294499 * A294501 A294502 A294503

Keyword

nonn

Author

Vaclav Kotesovec, Nov 01 2017

Status

approved