A005494 3-Bell numbers: E.g.f.: exp(3*z + exp(z) - 1).
1, 4, 17, 77, 372, 1915, 10481, 60814, 372939, 2409837, 16360786, 116393205, 865549453, 6713065156, 54190360453, 454442481041, 3952241526188, 35590085232519, 331362825860749, 3185554606447814, 31581598272055879, 322516283206446897, 3389017736055752178
Offset: 0
Keywords
Examples
G.f. = 1 + 4*x + 17*x^2 + 77*x^3 + 372*x^4 + 1915*x^5 + 10481*x^6 + 60814*x^7 + ...
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
- 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.
- Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
- I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.
- J. Riordan, Letter, Oct 31 1977
- N. J. A. Sloane, Transforms
- Earl Glen Whitehead Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.
Crossrefs
Programs
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Magma
A005494:= func< n | (&+[Binomial(n,j)*3^(n-j)*Bell(j): j in [0..n]]) >; [A005494(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022
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Maple
seq(add(3^(n-i)*combinat:-bell(i)*binomial(n,i),i=0..n), n=0..50); # Robert Israel, Dec 16 2014 # second Maple program: b:= proc(n, m) option remember; `if`(n=0, m^2, m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> b(n+1, 0)-b(n, 0): seq(a(n), n=0..23); # Alois P. Heinz, Aug 03 2025
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Mathematica
Range[0, 40]! CoefficientList[Series[Exp[3 x + Exp[x] - 1], {x, 0, 40}], x] (* Vincenzo Librandi, Mar 04 2014 *) -
SageMath
def A005494(n): return sum( 3^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1)) [A005494(n) for n in range(31)] # G. C. Greubel, Dec 01 2022
Formula
a(n) = Sum_{i=0..n} 3^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007
a(n) = exp(-1)*Sum_{k>=0} ((k+3)^n)/k!. - Gerald McGarvey, Jun 03 2004. May be rewritten as a(n) = Sum_{k>=3} (k^n*(k-1)*(k-2)/k!)/exp(1), which is a Dobinski-type relation for this sequence. - Karol A. Penson, Aug 18 2006
Define f_1(x), f_2(x), ... such that f_1(x) = x^2*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,-3). - Milan Janjic, Jul 08 2010
a(n) = Sum_{k=3..n+3} A143495(n+3,k), n >= 0. - Wolfdieter Lang, Sep 29 2011
G.f.: 1/U(0) where U(k)= 1 - x*(k+4) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: Sum_{k>0} x^(k-1) / ((1 - 3*x) * (1 - 4*x) * ... * (1 - (k+2)*x)). - Michael Somos, Feb 26 2014
G.f.: Sum_{k>0} k * x^(k-1) / ((1 - 2*x) * (1 - 3*x) * ... * (1 - (k+1)*x)). - Michael Somos, Feb 26 2014
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 3) / LambertW(n)^(n + 7/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{k=0..n} 4^k*A124323(n, k). - Mélika Tebni, Jun 10 2022
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