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Expansion of e.g.f. sinh(sin(x)^2) (even powers only).
3

%I #24 Feb 01 2018 03:44:15

%S 0,2,-8,152,-6848,312992,-17468288,1385712512,-143250864128,

%T 17321985331712,-2418583050119168,396575082357512192,

%U -75839817371814084608,16517549362496506929152

%N Expansion of e.g.f. sinh(sin(x)^2) (even powers only).

%H G. C. Greubel, <a href="/A009596/b009596.txt">Table of n, a(n) for n = 0..250</a>

%F a(n) = 8*Sum_{j=1..n} 2^(2*n-4*j)/(2*j-1)!*Sum_{i=0..2*j-1} (i-2*j+1)^(2*n)*binomial(4*j-2,i)*(-1)^(n-1-i). - _Vladimir Kruchinin_, Jun 08 2011

%e sinh(sin(x)*sin(x)) = 2/2!*x^2 - 8/4!*x^4 + 152/6!*x^6 - 6848/8!*x^8... - Patrick Demichel (patrick.demichel(AT)hp.com)

%t With[{nmax = 50}, CoefficientList[Series[Sinh[Sin[x]^2], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]] (* _G. C. Greubel_, Jan 30 2018 *)

%o (Maxima)

%o a(n):=8*sum(2^(2*n-4*j)/(2*j-1)!*sum((i-2*j+1)^(2*n)*binomial(4*j-2,i)*(-1)^(n-1-i),i,0,2*j-1),j,1,n); /* _Vladimir Kruchinin_, Jun 08 2011 */

%o (PARI) x='x+O('x^50); v=Vec(serlaplace(sinh(sin(x)^2))); concat([0], vector(#v\2,n,v[2*n-1])) \\ _G. C. Greubel_, Jan 30 2018

%K sign

%O 0,2

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997