A129695 Laguerre transform of the Jacobsthal numbers.

0, 1, 5, 30, 221, 1936, 19587, 223924, 2846741, 39763152, 604552571, 9929914204, 175116159429, 3298466345656, 66063837734819, 1401515958032628, 31386104948551253, 739730654456796832, 18299498906318500683, 474007927812558263308, 12828197342517251892485

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..440

Formula

a(n) = Sum_{k=0..n} C(n,k)*n!*A001045(k)/k!.

a(n) = (n!/3)*( LaguerreL(n,-2) - LaguerreL(n,1) ).

Conjecture: a(n) +(-4*n+3)*a(n-1) +(6*n^2-16*n+9)*a(n-2) -(4*n-7)*(n-2)^2*a(n-3) +(n-2)^2*(n-3)^2*a(n-4)=0. - R. J. Mathar, Feb 23 2015

a(n) ~ n^(n + 1/4) / (3*2^(3/4) * exp(n-2*sqrt(2*n)+1)) * (1 + 67/(48*sqrt(2*n))). - Vaclav Kotesovec, Nov 13 2017

Maple

A129695 := proc(n)
    add(binomial(n, k)*n!*A001045(k)/k!, k=0..n) ;
end proc: # R. J. Mathar, Feb 23 2015

Mathematica

Table[n!*(LaguerreL[n, -2] - LaguerreL[n, 1])/3, {n, 0, 20}] (* Vaclav Kotesovec, Nov 13 2017 *)
a[n_] := Sum[n!*Binomial[n, k]*((2^k -(-1)^k)/3)/k!, {k, 0, n}]; Table[a[n], {n, 0, 40}] (* G. C. Greubel, May 14 2018 *)

Prog

(PARI) for(n=0, 30, print1(sum(k=0, n, n!*binomial(n, k)*((2^k -(-1)^k)/3)/k!), ", ")) \\ G. C. Greubel, May 14 2018
(Magma) [(&+[Factorial(n)*Binomial(n, k)*((2^k -(-1)^k)/3)/Factorial(k) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 14 2018

Crossrefs

Cf. A001045, A105277.

Sequence in context: A058247 A137965 A371544 * A110521 A318920 A363908

Adjacent sequences: A129692 A129693 A129694 * A129696 A129697 A129698

Keyword

easy,nonn

Author

Paul Barry, May 01 2007

Extensions

Terms a(17) onward added by G. C. Greubel, May 14 2018

Status

approved