A078945 Row sums of A078939.

1, 5, 29, 189, 1357, 10589, 88909, 797085, 7583373, 76179037, 804638925, 8904557341, 102929260813, 1239432543709, 15511264432973, 201330839371421, 2705249923950477, 37567754666530141, 538369104335121869

Offset: 0

Author

Paul D. Hanna, Dec 18 2002

Keywords

• nonn

Comments

Equals A078944(n+1)/4.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Column k=4 of A335975.

Cf. A078939, A078944, A000110, A035009, A078940.

Programs

Maple

A078945 := proc(n) local a,b,i;
a := [seq(2,i=1..n)]; b := [seq(1,i=1..n)];
exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=4,%),66)) end:
seq(A078945(n),n=0..18); # Peter Luschny, Mar 30 2011

Mathematica

Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4+x), {x, 0, 20}], x]
Table[1/E^4/4*Sum[m^n/m!*4^m,{m,0,Infinity}],{n,1,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 4]/4, {n, 0, 20}] (* Vaclav Kotesovec, Jun 26 2022 *)

Formula

E.g.f.: exp(4*(exp(x)-1)+x).

Stirling transform of [1, 4, 4^2, 4^3, ...]. - Gerald McGarvey, Jun 01 2005

Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n)=e^{-4}*f_n(4). - Milan Janjic, May 30 2008

G.f.: 1/(Q(0) - 4*x) where Q(k) = 1 - x*(k+1)/( 1 - 4*x/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 22 2013

G.f.: T(0)/(1-5*x), where T(k) = 1 - 4*x^2*(k+1)/( 4*x^2*(k+1) - (1-5*x-x*k)*(1-6*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013

a(n) = exp(-4) * Sum_{k>=0} (k + 1)^n * 4^k / k!. - Ilya Gutkovskiy, Apr 20 2020

a(n) ~ n^(n+1) * exp(n/LambertW(n/4) - n - 4) / (4 * sqrt(1 + LambertW(n/4)) * LambertW(n/4)^(n+1)). - Vaclav Kotesovec, Jun 26 2022

a(0) = 1; a(n) = a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Dec 05 2023

Extensions

More terms from Robert G. Wilson v, Dec 19 2002