OFFSET
0,3
LINKS
Robert Israel, Table of n, a(n) for n = 0..449
Eric Weisstein's MathWorld, Incomplete Gamma Function
FORMULA
a(n) = (Gamma(n+1, sqrt(2))*exp(sqrt(2)) - Gamma(n+1, -sqrt(2))/exp(sqrt(2))) / (2*sqrt(2)).
a(n) ~ sqrt(Pi)*sinh(sqrt(2))*n^(n+1/2)*exp(-n).
D-finite with recurrence: a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3).
Gamma(n+1, sqrt(2))*exp(sqrt(2)) = A277431(n) + sqrt(2)*a(n).
Gamma(n+1, -sqrt(2))/exp(sqrt(2)) = A277431(n) - sqrt(2)*a(n).
For n > 0, a(2*n) = 2*n*a(2*n-1).
MAPLE
f:= gfun:-rectoproc({a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3), a(0)=0, a(1)=1, a(2)=2}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Oct 30 2016
MATHEMATICA
Round@Table[(Gamma[n + 1, Sqrt[2]] Exp[Sqrt[2]] - Gamma[n + 1, -Sqrt[2]]/Exp[Sqrt[2]])/(2 Sqrt[2]), {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
Expand@Table[SeriesCoefficient[Sinh[Sqrt[2] x]/(Sqrt[2] (1 - x)), {x, 0, n}] n!, {n, 0, 20}]
PROG
(PARI) x='x+O('x^30); concat([0], round(Vec(serlaplace(sinh(sqrt(2)*x)/( sqrt(2)*(1-x)))))) \\ G. C. Greubel, Aug 19 2018
(Magma) I:=[1, 2, 8]; [0] cat [n le 3 select I[n] else n*Self(n-1) + 2*Self(n-2) - 2*(n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 19 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Reshetnikov, Oct 14 2016
STATUS
approved