A129695 - OEIS

%I #13 Sep 08 2022 08:45:30

%S 0,1,5,30,221,1936,19587,223924,2846741,39763152,604552571,9929914204,

%T 175116159429,3298466345656,66063837734819,1401515958032628,

%U 31386104948551253,739730654456796832,18299498906318500683,474007927812558263308,12828197342517251892485

%N Laguerre transform of the Jacobsthal numbers.

%H G. C. Greubel, <a href="/A129695/b129695.txt">Table of n, a(n) for n = 0..440</a>

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

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

%F 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

%F 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

%p A129695 := proc(n)

%p     add(binomial(n,k)*n!*A001045(k)/k!,k=0..n) ;

%p end proc: # _R. J. Mathar_, Feb 23 2015

%t Table[n!*(LaguerreL[n, -2] - LaguerreL[n, 1])/3, {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 13 2017 *)

%t 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 *)

%o (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

%o (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

%Y Cf. A001045, A105277.

%K easy,nonn

%O 0,3

%A _Paul Barry_, May 01 2007

%E Terms a(17) onward added by _G. C. Greubel_, May 14 2018