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

%I A277432 #24 Feb 16 2025 08:33:36
%S A277432 0,1,2,8,32,164,984,6896,55168,496528,4965280,54618112,655417344,
%T A277432 8520425536,119285957504,1789289362688,28628629803008,486686706651392,
%U A277432 8760360719725056,166446853674776576,3328937073495531520,69907678543406162944,1537968927954935584768
%N A277432 E.g.f.: sinh(sqrt(2)*x)/(sqrt(2)*(1-x)).
%H A277432 Robert Israel, <a href="/A277432/b277432.txt">Table of n, a(n) for n = 0..449</a>
%H A277432 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>
%F A277432 a(n) = (Gamma(n+1, sqrt(2))*exp(sqrt(2)) - Gamma(n+1, -sqrt(2))/exp(sqrt(2))) / (2*sqrt(2)).
%F A277432 a(n) ~ sqrt(Pi)*sinh(sqrt(2))*n^(n+1/2)*exp(-n).
%F A277432 D-finite with recurrence: a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3).
%F A277432 Gamma(n+1, sqrt(2))*exp(sqrt(2)) = A277431(n) + sqrt(2)*a(n).
%F A277432 Gamma(n+1, -sqrt(2))/exp(sqrt(2)) = A277431(n) - sqrt(2)*a(n).
%F A277432 For n > 0, a(2*n) = 2*n*a(2*n-1).
%p A277432 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):
%p A277432 map(f, [$0..20]); # _Robert Israel_, Oct 30 2016
%t A277432 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 *)
%t A277432 Expand@Table[SeriesCoefficient[Sinh[Sqrt[2] x]/(Sqrt[2] (1 - x)), {x, 0, n}] n!, {n, 0, 20}]
%o A277432 (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
%o A277432 (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
%Y A277432 Cf. A000023, A263823, A277345, A277431.
%K A277432 nonn
%O A277432 0,3
%A A277432 _Vladimir Reshetnikov_, Oct 14 2016