cp's OEIS Frontend

A218307 E.g.f. A(x) satisfies A( x/(exp(4*x)*cosh(4*x)) ) = exp(x)*cosh(x).

%I A218307 #10 Apr 24 2024 08:55:17
%S A218307 1,1,10,244,9288,483216,31949216,2564959552,242374510720,
%T A218307 26355555496192,3241906046249472,445085008158569472,
%U A218307 67469834196870809600,11192986206960277688320,2017105871358529382883328,392394481517424330142203904,81955683182673295403291541504
%N A218307 E.g.f. A(x) satisfies A( x/(exp(4*x)*cosh(4*x)) ) = exp(x)*cosh(x).
%C A218307 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218307 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%F A218307 E.g.f.: A(x) = Sum_{n>=0} (4*n+1)^(n-1) * cosh((4*n+1)*x) * x^n/n!.
%F A218307 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218307 E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/4 * LambertW(-4*x * exp(4*x)) ).
%F A218307 a(n) = 1/2 * Sum_{k=0..n} (4*k+1)^(n-1) * binomial(n,k) for n > 0.
%F A218307 G.f.: 1/2 + 1/2 * Sum_{k>=0} (4*k+1)^(k-1) * x^k/(1 - (4*k+1)*x)^(k+1). (End)
%e A218307 E.g.f.: A(x) = 1 + x + 10*x^2/2! + 244*x^3/3! + 9288*x^4/4! + 483216*x^5/5! +...
%e A218307 where
%e A218307 A(x) = cosh(x) + 5^0*cosh(5*x)*x + 9^1*cosh(9*x)*x^2/2! + 13^2*cosh(13*x)*x^3/3! + 17^3*cosh(17*x)*x^4/4! + 21^4*cosh(21*x)*x^5/5! +...
%o A218307 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(4*X)*cosh(4*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
%o A218307 for(n=0,25,print1(a(n),", "))
%o A218307 (PARI) /* Formula derived from a LambertW identity: */
%o A218307 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(4*k+1)^(k-1)*cosh((4*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218307 for(n=0,25,print1(a(n),", "))
%Y A218307 Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218308, A218309, A218310.
%K A218307 nonn
%O A218307 0,3
%A A218307 _Paul D. Hanna_, Oct 25 2012