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A001907 Expansion of e^(-x)/(1-4x).
(Formerly M3112 N1261)
5
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

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.

Sequence in context: A229162 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

STATUS

approved

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Last modified February 24 14:41 EST 2018. Contains 299623 sequences. (Running on oeis4.)