A277432 E.g.f.: sinh(sqrt(2)*x)/(sqrt(2)*(1-x)).

0, 1, 2, 8, 32, 164, 984, 6896, 55168, 496528, 4965280, 54618112, 655417344, 8520425536, 119285957504, 1789289362688, 28628629803008, 486686706651392, 8760360719725056, 166446853674776576, 3328937073495531520, 69907678543406162944, 1537968927954935584768

Offset: 0

Author

Vladimir Reshetnikov, Oct 14 2016

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..449
• Eric Weisstein's MathWorld, Incomplete Gamma Function

Crossrefs

Cf. A000023, A263823, A277345, A277431.

Programs

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

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}]

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

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).