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

1, 1, 4, 25, 217, 2416, 32839, 527185, 9761602, 204800551, 4801461049, 124402647370, 3529848676237, 108859319101261, 3625569585663484, 129689000146431205, 4958830249864725997, 201834650901695603296, 8712774828941647677019, 397596632650906687905565

Offset

0,3

Comments

Stirling transform of A001147.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..391

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Stirling Transform

Formula

E.g.f.: 1/sqrt(3 - 2*exp(x)).

a(n) = Sum_{k=0..n} Stirling2(n,k)*(2*k - 1)!!.

a(n) ~ sqrt(2/3) * n^n / ((log(3/2))^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jul 01 2018

Conjecture: a(n) = Sum_{k>=0} k^n * binomial(2*k,k) / (2^k * 3^(k + 1/2)). - Diego Rattaggi, Oct 11 2020

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

a(0) = 1; a(n) = Sum_{k=1..n} (2 - k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

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

Maple

b:= proc(n, m) option remember;
     `if`(n=0, doublefactorial(2*m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 04 2021

Mathematica

nmax = 19; CoefficientList[Series[Sum[(2 k - 1)!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[1/Sqrt[3 - 2 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] (2 k - 1)!!, {k, 0, n}], {n, 0, 19}]

Crossrefs

Cf. A000670, A001147, A004123, A305405, A346982 - A346985.

Sequence in context: A351767 A198198 A007830 * A218826 A060911 A060912

Adjacent sequences: A305401 A305402 A305403 * A305405 A305406 A305407

Keyword

nonn,easy

Author

Ilya Gutkovskiy, May 31 2018

Status

approved