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%I #6 Mar 11 2022 02:11:13
%S 1,-2,10,-62,250,3538,-109430,376738,64406170,-1496149262,
%T -66387156950,4120939699138,114360544465210,-16447057086702062,
%U -315993884108535350,99921676927889325538,1478937314465295441370,-907773678752741550637262,-14225447208333541085396630
%N a(0) = 1; a(n) = -2 * Sum_{k=1..n} binomial(2*n-1,2*k-1) * a(n-k).
%F E.g.f.: exp( 2 * (1 - cosh(x)) ) (even powers only).
%t a[0] = 1; a[n_] := a[n] = -2 Sum[Binomial[2 n - 1, 2 k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%t nmax = 36; Take[CoefficientList[Series[Exp[2 (1 - Cosh[x])], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
%Y Cf. A000807, A210657, A260884, A352278.
%K sign
%O 0,2
%A _Ilya Gutkovskiy_, Mar 10 2022