cp's OEIS Frontend

A305404 Expansion of Sum_{k>=0} (2*k - 1)!!*x^k/Product_{j=1..k} (1 - j*x).

%I A305404 #29 Feb 16 2025 08:33:54
%S A305404 1,1,4,25,217,2416,32839,527185,9761602,204800551,4801461049,
%T A305404 124402647370,3529848676237,108859319101261,3625569585663484,
%U A305404 129689000146431205,4958830249864725997,201834650901695603296,8712774828941647677019,397596632650906687905565
%N A305404 Expansion of Sum_{k>=0} (2*k - 1)!!*x^k/Product_{j=1..k} (1 - j*x).
%C A305404 Stirling transform of A001147.
%H A305404 Alois P. Heinz, <a href="/A305404/b305404.txt">Table of n, a(n) for n = 0..391</a>
%H A305404 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305404 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A305404 E.g.f.: 1/sqrt(3 - 2*exp(x)).
%F A305404 a(n) = Sum_{k=0..n} Stirling2(n,k)*(2*k - 1)!!.
%F A305404 a(n) ~ sqrt(2/3) * n^n / ((log(3/2))^(n + 1/2) * exp(n)). - _Vaclav Kotesovec_, Jul 01 2018
%F A305404 Conjecture: a(n) = Sum_{k>=0} k^n * binomial(2*k,k) / (2^k * 3^(k + 1/2)). - _Diego Rattaggi_, Oct 11 2020
%F A305404 O.g.f. conjectural: 1/(1 - x/(1 - 3*x/(1 - 3*x/(1 - 6*x/(1 - 5*x/(1 - 9*x/(1 - 7*x/(1 - ... - (2*n-1)*x/(1 - 3*n*x/(1 - ... )))))))))) - a continued fraction of Stieltjes-type. - _Peter Bala_, Dec 06 2020
%F A305404 a(0) = 1; a(n) = Sum_{k=1..n} (2 - k/n) * binomial(n,k) * a(n-k). - _Seiichi Manyama_, Sep 09 2023
%F A305404 a(0) = 1; a(n) = a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 16 2023
%p A305404 b:= proc(n, m) option remember;
%p A305404      `if`(n=0, doublefactorial(2*m-1), m*b(n-1, m)+b(n-1, m+1))
%p A305404     end:
%p A305404 a:= n-> b(n, 0):
%p A305404 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 04 2021
%t A305404 nmax = 19; CoefficientList[Series[Sum[(2 k - 1)!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A305404 nmax = 19; CoefficientList[Series[1/Sqrt[3 - 2 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
%t A305404 Table[Sum[StirlingS2[n, k] (2 k - 1)!!, {k, 0, n}], {n, 0, 19}]
%Y A305404 Cf. A000670, A001147, A004123, A305405, A346982 - A346985.
%K A305404 nonn,easy
%O A305404 0,3
%A A305404 _Ilya Gutkovskiy_, May 31 2018