A266232 Binomial transform of the number of partitions into distinct parts (A000009).

1, 2, 4, 9, 21, 49, 114, 265, 615, 1422, 3272, 7493, 17090, 38850, 88065, 199097, 448953, 1009788, 2265642, 5071611, 11328395, 25254093, 56195143, 124829822, 276839061, 612991848, 1355268779, 2992016128, 6596222234, 14522634554, 31933047707, 70130243427

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

Cf. A000009, A294467, A293467, A294468.

Cf. A218481, A294466, A281425, A095051, A294500.

Sequence in context: A281425 A101891 A119967 * A052921 A219150 A322325

Adjacent sequences: A266229 A266230 A266231 * A266233 A266234 A266235

Keyword

nonn

Author

Vaclav Kotesovec, Dec 25 2015

Status

approved