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A005119
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Infinitesimal generator of x(x+1).
(Formerly M3024)
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2
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1, 1, 3, 16, 124, 1256, 15576, 226248, 3729216, 68179968, 1361836800, 29501349120, 693638208000, 17815908096000, 502048890201600, 15388268595840000, 500579319427891200, 16817771937344716800, 581609175119297740800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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REFERENCES
| G. Labelle, Sur l'inversion et l'iteration continue des series formelles, Europ. J. Combin., 1 (1980), 113-138.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| a(n) = (n-2)!*Sum_{i=1..n-1} (-1)^(i+1)*C(n-i+1,i+1)*a(n-i)/(n-i-1)! for n>1 with a(1)=1. E.g.f. satisfies: A(x) = (1-x)^2/(1-2x)*A(x-x^2) where A(x) = Sum_{n>=0}a(n+1)*x^n/n! with offset so that A(0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 27 2007
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MATHEMATICA
| max = 19; f[x_] := Sum[a[n+1]*x^n/n!, {n, 0, max}]; coes = CoefficientList[ Series[ f[x]-((1-x)^2/(1-2*x))*f[x-x^2], {x, 0, max}], x]; Array[a, max] /. Solve[a[1] == a[2] == 1 && Thread[coes == 0]][[1]] (* From Jean-François Alcover, Nov 03 2011 *)
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PROG
| (PARI) {a(n)=if(n<1, 0, if(n==1, 1, (n-2)!*sum(i=1, n-1, (-1)^(i+1)*binomial(n-i+1, i+1)*a(n-i)/(n-i-1)!)))} - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 27 2007
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CROSSREFS
| Sequence in context: A035352 A159607 A087018 * A190291 A090135 A188417
Adjacent sequences: A005116 A005117 A005118 * A005120 A005121 A005122
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KEYWORD
| nonn,nice
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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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EXTENSIONS
| More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Dec 27 2007
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