cp's OEIS Frontend

A218301 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(3*x)*cosh(3*x).

%I A218301 #17 Feb 16 2025 08:33:18
%S A218301 1,3,24,252,3360,55008,1074816,24499968,639744000,18856765440,
%T A218301 619897847808,22502300590080,894419152404480,38651030120693760,
%U A218301 1804765006764441600,90574514900736933888,4862862027933962207232,278158492957848901779456,16889663645642083220324352
%N A218301 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(3*x)*cosh(3*x).
%C A218301 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218301 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%H A218301 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A218301 E.g.f.: A(x) = Sum_{n>=0} 3*(n+3)^(n-1) * cosh((n+3)*x) * x^n/n!.
%F A218301 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218301 E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3*LambertW(-x * exp(x)) ).
%F A218301 a(n) = 3/2 * Sum_{k=0..n} (k+3)^(n-1) * binomial(n,k) for n > 0.
%F A218301 G.f.: 1/2 + 3/2 * Sum_{k>=0} (k+3)^(k-1) * x^k/(1 - (k+3)*x)^(k+1). (End)
%e A218301 E.g.f.: A(x) = 1 + 3*x + 24*x^2/2! + 252*x^3/3! + 3360*x^4/4! + 55008*x^5/5! +...
%e A218301 where
%e A218301 A(x) = cosh(3*x) + 3*4^0*cosh(4*x)*x + 3*5^1*cosh(5*x)*x^2/2! + 3*6^2*cosh(6*x)*x^3/3! + 3*7^3*cosh(7*x)*x^4/4! + 3*8^4*cosh(8*x)*x^5/5! +...
%o A218301 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(X)*cosh(X)))); Egf=exp(3*R)*cosh(3*R); n!*polcoeff(Egf,n)}
%o A218301 for(n=0,25,print1(a(n),", "))
%o A218301 (PARI) /* Formula derived from a LambertW identity: */
%o A218301 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,3*(k+3)^(k-1)*cosh((k+3)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218301 for(n=0,25,print1(a(n),", "))
%Y A218301 Cf. A201595, A218300, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310.
%K A218301 nonn
%O A218301 0,2
%A A218301 _Paul D. Hanna_, Oct 25 2012