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 A001907 Expansion of e.g.f. exp(-x)/(1-4*x). (Formerly M3112 N1261) 9
 1, 3, 25, 299, 4785, 95699, 2296777, 64309755, 2057912161, 74084837795, 2963393511801, 130389314519243, 6258687096923665, 325451729040030579, 18225296826241712425, 1093517809574502745499, 69985139812768175711937, 4758989507268235948411715 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES 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). LINKS Harvey P. Dale, Table of n, a(n) for n = 0..350 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1. FORMULA 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 a(n) ~ n! * exp(-1/4)*4^n. - 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 MAPLE 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): map(f, [\$0..30]); # Robert Israel, Aug 07 2015 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A000166, A000354, A000180, A001908. Column k=4 of A320032. Sequence in context: A347255 A292111 A123989 * A212722 A236268 A181085 Adjacent sequences: A001904 A001905 A001906 * A001908 A001909 A001910 KEYWORD easy,nonn AUTHOR N. J. A. Sloane EXTENSIONS More terms from Ralf Stephan, May 22 2004 Typo fixed by Charles R Greathouse IV, Oct 28 2009 Name clarified by Ilya Gutkovskiy, Jan 17 2020 STATUS approved

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Last modified July 19 19:39 EDT 2024. Contains 374436 sequences. (Running on oeis4.)