A078940 Row sums of A078938.

1, 4, 19, 103, 622, 4117, 29521, 227290, 1865881, 16239523, 149142952, 1439618143, 14555631781, 153700654036, 1690684883191, 19328770917499, 229203640111870, 2814018686591089, 35711716110387589, 467766675528462562

Offset: 0

Author

Paul D. Hanna, Dec 18 2002

Keywords

• nonn

Comments

Divide by 3^n and insert an initial 1 to get sequence that shifts left one place under 1/3 order binomial transformation. - Franklin T. Adams-Watters, Jul 13 2006

Binomial transform of A027710. - Vaclav Kotesovec, Jun 26 2022

Links

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

Crossrefs

Column k=3 of A335975.

Cf. A027710, A035009, A078938, A078945, A355254.

Programs

Maple

A078940 := 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=3,%),66)) end:
seq(A078940(n),n=0..19); # Peter Luschny, Mar 30 2011

Mathematica

Table[n!, {n, 0, 20}]CoefficientList[Series[E^(3E^x-3+x), {x, 0, 20}], x]
Table[1/E^3/3*Sum[m^n/m!*3^m,{m,0,Infinity}],{n,1,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 3]/3, {n, 0, 20}] (* Vaclav Kotesovec, Jan 15 2016 *)
nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - (k+4)*x - 3*(k+1)*x^2/g[k+1]; CoefficientList[Series[1/g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 15 2016, after Sergei N. Gladkovskii *)

Formula

E.g.f.: exp(3*(exp(x)-1)+x).

Stirling transform of [1, 3, 3^2, 3^3, ...]. - 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^{-3}*f_n(3). - Milan Janjic, May 30 2008

G.f.: 1/T(0), where T(k) = 1 - (k+4)*x - 3*(k+1)*x^2/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2016

a(n) = exp(-3) * Sum_{k>=0} (k + 1)^n * 3^k / k!. - Ilya Gutkovskiy, Apr 20 2020

a(n) ~ n^(n+1) * exp(n/LambertW(n/3) - n - 3) / (3 * sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n+1)). - Vaclav Kotesovec, Jun 26 2022

a(0) = 1; a(n) = a(n-1) + 3 * 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