A255927 a(n) = (3/4) * Sum_{k>=0} (3*k)^n/4^k.

1, 1, 5, 33, 285, 3081, 40005, 606033, 10491885, 204343641, 4422082005, 105265315233, 2733583519485, 76902684021801, 2329889536156005, 75629701786875633, 2618654297178083085, 96336948993312237561, 3752590641305604502005, 154294551397830418471233, 6677999524135208461382685

Offset: 0

Author

Karol A. Penson, Sep 03 2015

Keywords

• nonn

Examples

			a(5) = 729*hypergeom([2,2,2,2,2],[1,1,1,1],1/4)/16 = 3081.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the Log-normal distribution, arXiv:quant-ph/0303030, 2003.
• P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the Log-normal distribution, J. Phys. A: Math. Gen. 36, (2003), L273.
• Eric Weisstein's World of Mathematics, Lerch Transcendent

Crossrefs

Cf. A000670, A094418, A004123 A032033, A094417, A122704, A326323.

Programs

Maple

S:= series(3/(4-exp(3*x)), x, 51):
seq(coeff(S,x,n)*n!, n=0..50); # Robert Israel, Sep 03 2015
seq(add(combinat:-eulerian1(n,k)*4^k, k=0..n), n=0..20); # Peter Luschny, Jun 27 2019

Mathematica

a[n_] := 3^(n+1)/4 HurwitzLerchPhi[1/4, -n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 18 2018 *)
Eulerian1[0, 0] = 1; Eulerian1[n_, k_] := Sum[(-1)^j (k-j+1)^n Binomial[n+1, j], {j, 0, k+1}]; Table[Sum[Eulerian1[n, k] 4^k, {k, 0, n}], {n, 0, 20}] (* Jean-François Alcover, Jul 13 2019, after Peter Luschny *)

PARI

a(n) = sum(k=0, n, stirling(n,k,2)*k!*3^(n-k)); \\ Michel Marcus, Sep 03 2015

Formula

a(n) = Sum_{k>=0} Stirling2(n,k)*k!*3^(n-k).

E.g.f.: 3/(4-exp(3*x)).

Special values of the generalized hypergeometric function n_F_(n-1):

a(n) = (3^(n+1)/16) * hypergeom([2,2,..2],[1,1,..1],1/4), where the sequence in the first square bracket ("upper" parameters) has n elements all equal to 2 whereas the sequence in the second square bracket ("lower" parameters) has n-1 elements all equal to 1.

Example: a(5) = 729 * hypergeom([2,2,2,2,2],[1,1,1,1],1/4)/16 = 3081.

a(n) is the n-th moment of the discrete weight function W(x) = (3/4)*sum(k>=0, Dirac(x-3*k)/4^k), n>=1.

a(n) ~ n! * 3^(n+1) / ((log(2))^(n+1) * 2^(n+3)). - Vaclav Kotesovec, Jul 09 2018

G.f.: Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - 3*k*x). - Ilya Gutkovskiy, Apr 04 2019

a(n) = A_{4}(n) where A_{n}(x) are the Eulerian polynomials as defined in A326323. - Peter Luschny, Jun 27 2019

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 18 2018