A045379 Expansion of e.g.f.: exp(4*z + exp(z) - 1).

1, 5, 26, 141, 799, 4736, 29371, 190497, 1291020, 9131275, 67310847, 516369838, 4116416797, 34051164985, 291871399682, 2588914083065, 23733360653955, 224592570163192, 2191466128865567, 22024934452712437, 227771488390279260

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.

Formula

a(n) = exp(-1)*Sum_{k>=0} ((k+4)^n)/k!. - Gerald McGarvey, Jun 03 2004

A recursive formula to compute some integer sequences (including A000110, A005493, A005494 and the present sequence). Define G(n, m), where n, m >= 0, as follows: G(0, m) = 1; G(n, m) = G(n-1, m) * (m+1) + G(n-1, m+1), where n > 0. Then G(n, 0) = A000110(n+1); G(n, 1) = A005493(n+1); G(n, 2) = A005494(n+1); G(n, 3) = A045379(n+1). - Alexey Andreev (ava12(AT)nm.ru), Jan 05 2006

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

Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i <= j), and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = (-1)^(n)*charpoly(A,-4). - Milan Janjic, Jul 08 2010

G.f.: 1/U(0) where U(k) = 1 - x*(k+5) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012

a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 4) / LambertW(n)^(n + 9/2). - Vaclav Kotesovec, Jun 10 2020

a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020

a(n) = Sum_{j=0..n} binomial(n, j)*4^(n-j)*A000110(j). - G. C. Greubel, Dec 01 2022

Mathematica

a[0]= 1; a[n_]:= a[n]= 4*a[n-1] +Sum[Binomial[n-1, k]*a[k], {k, 0, n-1}]; Array[a, 21, 0] (* Amiram Eldar, Jul 03 2020 *)

Prog

(Magma)
A045379:= func< n | (&+[Binomial(n, j)*4^(n-j)*Bell(j): j in [0..n]]) >;
[A045379(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022
(SageMath)
def A045379(n): return sum( 4^(n-j)*bell_number(j)*binomial(n, j) for j in range(n+1))
[A045379(n) for n in range(31)] # G. C. Greubel, Dec 01 2022

Crossrefs

Cf. A000110, A005493, A005494, A108087, A196834.

Equals the row sums of triangle A143496. - Wolfdieter Lang, Sep 29 2011

Sequence in context: A182401 A363308 A104498 * A053487 A277957 A183161

Adjacent sequences: A045376 A045377 A045378 * A045380 A045381 A045382

Keyword

nonn

Author

N. J. A. Sloane

Status

approved