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A005494
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E.g.f.: exp(3*z+exp(z)-1).
(Formerly M3540)
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8
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1, 4, 17, 77, 372, 1915, 10481, 60814, 372939, 2409837, 16360786, 116393205, 865549453, 6713065156, 54190360453, 454442481041, 3952241526188, 35590085232519, 331362825860749, 3185554606447814, 31581598272055879, 322516283206446897
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| From expansion of falling factorials (binomial transform of A005493).
a(n)=sum(A143495(n+3,k),k=3..n+3),n>=0. Row sums of Sheffer triangle (exp(3*x),exp(x)-1). [Wolfdieter Lang, Sep 29 2011]
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REFERENCES
| Toufik Mansour and Mark Shattuck, A RECURRENCE RELATED TO THE BELL NUMBERS, INTEGERS 11 (2011), #A67; http://www.emis.de/journals/INTEGERS/papers/l67/l67.pdf.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.
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LINKS
| J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
N. J. A. Sloane, Transforms
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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 (Gerald.McGarvey(AT)comcast.net), Jun 03 2004. May be rewritten as a(n)=sum(k^n*(k-1)*(k-2)/k!,k=3..infinity)/exp(1), which is a Dobinski-type relation for this sequence. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Aug 18 2006
Define f_1(x),f_2(x),... such that f_1(x)=x^2*e^x, f_{n+1}(x)=diff(x*f_n(x),x), for n=2,3,.... Then a(n-1)=e^{-1}*f_n(1). - Milan R. Janjic (agnus(AT)blic.net), 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). [From Milan R. Janjic (agnus(AT)blic.net), Jul 08 2010]
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CROSSREFS
| Cf. A000110, A005493.
Sequence in context: A151248 A104455 A123952 * A193782 A053486 A151249
Adjacent sequences: A005491 A005492 A005493 * A005495 A005496 A005497
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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