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A001907
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Expansion of e.g.f. exp(-x)/(1-4*x).
(Formerly M3112 N1261)
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9
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1, 3, 25, 299, 4785, 95699, 2296777, 64309755, 2057912161, 74084837795, 2963393511801, 130389314519243, 6258687096923665, 325451729040030579, 18225296826241712425, 1093517809574502745499, 69985139812768175711937, 4758989507268235948411715
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OFFSET
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0,2
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n,k)*k!*4^k. - Ralf Stephan, May 22 2004
Recurrence: a(n) = (4*n-1)*a(n-1) + 4*(n-1)*a(n-2). - Vaclav Kotesovec, Aug 16 2013
E.g.f. A(x) = exp(-x)/(1-4x) satisfies (1-4x)A' - (3+4x)A = 0. - Gheorghe Coserea, Aug 06 2015
a(n) = exp(-1/4)*4^n*Gamma(n+1,-1/4), where Gamma is the incomplete Gamma function. - Robert Israel, Aug 07 2015
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (4*k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
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MAPLE
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f:= gfun:-rectoproc({a(n) = (4*n-1)*a(n-1) + 4*(n-1)*a(n-2), a(0)=1, a(1)=3}, a(n), remember):
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[Exp[-x]/(1-4x), {x, 0, nn}], x] Range[0, nn]!] (* or *) Table[Sum[(-1)^(n+k) Binomial[n, k]k! 4^k, {k, 0, n}], {n, 0, 20}](* Harvey P. Dale, Oct 25 2011 *)
Join[{1}, RecurrenceTable[{a[1] == 3, a[2] == 25, a[n] == (4 n - 1) a[n-1] + 4(n - 1) a[n-2]}, a, {n, 20}]] (* Vincenzo Librandi, Aug 08 2015 *)
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PROG
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(PARI) a(n)=sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*4^k)
(PARI) x = 'x+O('x^33); Vec(serlaplace(exp(-x)/(1-4*x))) \\ Gheorghe Coserea, Aug 06 2015
(Magma) I:=[3, 25]; [1] cat [n le 2 select I[n] else (4*n-1)*Self(n-1)+4*(n-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 08 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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