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%I #13 Apr 24 2024 08:55:21
%S 1,2,20,392,11648,466112,23517824,1434077696,102618951680,
%T 8432793964544,782753794531328,81007725700038656,9249066952457584640,
%U 1154952975718091325440,156588371428134115868672,22908199202756436344963072,3597006040171205977538822144
%N E.g.f. A(x) satisfies A( x/(exp(3*x)*cosh(3*x)) ) = exp(2*x)*cosh(2*x).
%C More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x), then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%F E.g.f.: A(x) = Sum_{n>=0} 2*(3*n+2)^(n-1) * cosh((3*n+2)*x) * x^n/n!.
%F From _Seiichi Manyama_, Apr 23 2024: (Start)
%F E.g.f.: A(x) = 1/2 + 1/2 * exp( 2*x - 2/3 * LambertW(-3*x * exp(3*x)) ).
%F a(n) = Sum_{k=0..n} (3*k+2)^(n-1) * binomial(n,k) for n > 0.
%F G.f.: 1/2 + Sum_{k>=0} (3*k+2)^(k-1) * x^k/(1 - (3*k+2)*x)^(k+1). (End)
%e E.g.f.: A(x) = 1 + 2*x + 20*x^2/2! + 392*x^3/3! + 11648*x^4/4! + 466112*x^5/5! +...
%e where
%e A(x) = cosh(2*x) + 2*5^0*cosh(5*x)*x + 2*8^1*cosh(8*x)*x^2/2! + 2*11^2*cosh(11*x)*x^3/3! + 2*14^3*cosh(14*x)*x^4/4! + 2*17^4*cosh(17*x)*x^5/5! +...
%o (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(3*X)*cosh(3*X)))); Egf=exp(2*R)*cosh(2*R); n!*polcoeff(Egf,n)}
%o for(n=0,25,print1(a(n),", "))
%o (PARI) /* Formula derived from a LambertW identity: */
%o {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,2*(3*k+2)^(k-1)*cosh((3*k+2)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218307, A218308, A218309, A218310.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 25 2012
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