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%I #51 Dec 04 2022 08:33:18
%S 1,5,26,141,799,4736,29371,190497,1291020,9131275,67310847,516369838,
%T 4116416797,34051164985,291871399682,2588914083065,23733360653955,
%U 224592570163192,2191466128865567,22024934452712437,227771488390279260
%N Expansion of e.g.f.: exp(4*z + exp(z) - 1).
%H Seiichi Manyama, <a href="/A045379/b045379.txt">Table of n, a(n) for n = 0..500</a>
%H R. Jakimczuk, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Jakimczuk2/jakimczuk17.html">Successive Derivatives and Integer Sequences</a>, J. Int. Seq. 14 (2011) # 11.7.3.
%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Mezo/mezo9.html">The r-Bell numbers</a>, J. Int. Seq. 14 (2011) # 11.1.1.
%F a(n) = exp(-1)*Sum_{k>=0} ((k+4)^n)/k!. - _Gerald McGarvey_, Jun 03 2004
%F 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
%F 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
%F 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
%F 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
%F a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 4) / LambertW(n)^(n + 9/2). - _Vaclav Kotesovec_, Jun 10 2020
%F 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
%F a(n) = Sum_{j=0..n} binomial(n, j)*4^(n-j)*A000110(j). - _G. C. Greubel_, Dec 01 2022
%t 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 *)
%o (Magma)
%o A045379:= func< n | (&+[Binomial(n,j)*4^(n-j)*Bell(j): j in [0..n]]) >;
%o [A045379(n): n in [0..30]]; // _G. C. Greubel_, Dec 01 2022
%o (SageMath)
%o def A045379(n): return sum( 4^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1))
%o [A045379(n) for n in range(31)] # _G. C. Greubel_, Dec 01 2022
%Y Cf. A000110, A005493, A005494, A108087, A196834.
%Y Equals the row sums of triangle A143496. - _Wolfdieter Lang_, Sep 29 2011
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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