A001907 Expansion of e.g.f. exp(-x)/(1-4*x).

1, 3, 25, 299, 4785, 95699, 2296777, 64309755, 2057912161, 74084837795, 2963393511801, 130389314519243, 6258687096923665, 325451729040030579, 18225296826241712425, 1093517809574502745499, 69985139812768175711937, 4758989507268235948411715

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A000166, A000354, A000180, A001908.

Column k=4 of A320032.

Programs

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

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 *)

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

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

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