cp's OEIS Frontend

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

%I A266232 #23 Nov 02 2023 07:00:08
%S A266232 1,2,4,9,21,49,114,265,615,1422,3272,7493,17090,38850,88065,199097,
%T A266232 448953,1009788,2265642,5071611,11328395,25254093,56195143,124829822,
%U A266232 276839061,612991848,1355268779,2992016128,6596222234,14522634554,31933047707,70130243427
%N A266232 Binomial transform of the number of partitions into distinct parts (A000009).
%C A266232 Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
%C A266232 Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
%C A266232 g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
%C A266232 Special cases:
%C A266232 p < 1/2, g(n) = 0
%C A266232 p = 1/2, g(n) = r^2/16
%C A266232 p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
%C A266232 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 A266232 p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
%C A266232 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 A266232 Vaclav Kotesovec, <a href="/A266232/b266232.txt">Table of n, a(n) for n = 0..3200</a>
%F A266232 a(n) ~ 2^(n-5/4) * exp(Pi*sqrt(n/6) + Pi^2/48) / (3^(1/4)*n^(3/4)).
%F A266232 G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k/(1 - x)^k). - _Ilya Gutkovskiy_, Aug 19 2018
%t A266232 Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}]
%t A266232 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 *)
%Y A266232 Cf. A000009, A294467, A293467, A294468.
%Y A266232 Cf. A218481, A294466, A281425, A095051, A294500.
%K A266232 nonn
%O A266232 0,2
%A A266232 _Vaclav Kotesovec_, Dec 25 2015