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%I #8 Jan 29 2019 04:42:57
%S 0,1,1,1,-2,-7,6,119,120,-2911,-12518,90055,977164,-2167375,-83354634,
%T -168068473,7777602768,58283146817,-727882529102,-12779261480825,
%U 46543629605236,2663317412960849,7760606919565134,-548896641490323385,-5830401238269419400,104847450848773542497
%N Expansion of e.g.f. log(1 + exp(x)*sinh(sqrt(2)*x)/sqrt(2)).
%F E.g.f.: log(1 + Sum_{k>=1} Pell(k)*x^k/k!).
%F a(0) = 0; a(n) = Pell(n) - (1/n)*Sum_{k=1..n-1} binomial(n,k)*Pell(n-k)*k*a(k).
%p seq(n!*coeff(series(log(1+exp(x)*sinh(sqrt(2)*x)/sqrt(2)),x=0,26),x,n),n=0..25); # _Paolo P. Lava_, Jan 29 2019
%t FullSimplify[nmax = 25; CoefficientList[Series[Log[1 + Exp[x] Sinh[Sqrt[2] x]/Sqrt[2]], {x, 0, nmax}], x] Range[0, nmax]!]
%t a[n_] := a[n] = Fibonacci[n, 2] - Sum[Binomial[n, k] Fibonacci[n - k, 2] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 25}]
%Y Cf. A000129, A006673, A007553, A112005, A279271.
%K sign
%O 0,5
%A _Ilya Gutkovskiy_, Jan 25 2019
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