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
KEYWORD
easy,nonn
AUTHOR
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