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..3200
FORMULA
a(n) ~ 2^(n-5/4) * exp(Pi*sqrt(n/6) + Pi^2/48) / (3^(1/4)*n^(3/4)).
G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k/(1 - x)^k). - Ilya Gutkovskiy, Aug 19 2018
MATHEMATICA
Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}]
nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] * x^k / (1-x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Dec 25 2015
STATUS
approved
