A218309 - OEIS

%I #9 Apr 24 2024 08:48:22

%S 1,4,56,1264,40640,1711744,89533184,5607463936,409621790720,

%T 34218229227520,3219000547131392,336858779869020160,

%U 38823224436435845120,4886982191317154529280,667188807538423365632000,98200163047169655115350016,15501781660715229538766815232

%N E.g.f. A(x) satisfies A( x/(exp(3*x)*cosh(3*x)) ) = exp(4*x)*cosh(4*x).

%C More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

%C 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} 4*(3*n+4)^(n-1) * cosh((3*n+4)*x) * x^n/n!.

%F From _Seiichi Manyama_, Apr 23 2024: (Start)

%F E.g.f.: A(x) = 1/2 + 1/2 * exp( 4*x - 4/3 * LambertW(-3*x * exp(3*x)) ).

%F a(n) = 2 * Sum_{k=0..n} (3*k+4)^(n-1) * binomial(n,k) for n > 0.

%F G.f.: 1/2 + 2 * Sum_{k>=0} (3*k+4)^(k-1) * x^k/(1 - (3*k+4)*x)^(k+1). (End)

%e E.g.f.: A(x) = 1 + 4*x + 56*x^2/2! + 1264*x^3/3! + 40640*x^4/4! + 1711744*x^5/5! +...

%e where

%e A(x) = cosh(2*x) + 4*5^0*cosh(5*x)*x + 4*8^1*cosh(8*x)*x^2/2! + 4*11^2*cosh(11*x)*x^3/3! + 4*14^3*cosh(14*x)*x^4/4! + 4*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(4*R)*cosh(4*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,4*(3*k+4)^(k-1)*cosh((3*k+4)*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, A218306, A218307, A218308, A218310.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 25 2012