A258114 - OEIS

%I #15 Oct 19 2017 03:43:42

%S 1,1,2,9,72,665,6960,85057,1199744,19070865,336372480,6522635801,

%T 137996694528,3163206890857,78085740701696,2065239729737745,

%U 58263449436979200,1746433243580269217,55428341343200280576,1856918215298125692073,65483209810866254643200,2424691204935999655757241

%N E.g.f.: Sum_{n>=0} x^n * cosh(n*x).

%H G. C. Greubel, <a href="/A258114/b258114.txt">Table of n, a(n) for n = 0..410</a>

%F E.g.f.: (1 - x*cosh(x)) / (1 - 2*x*cosh(x) + x^2).

%F a(n) = Sum_{k=0..n} n!/k! * ((n-k)^k + (-n+k)^k)/2.

%F a(n) ~ n! * (1-c*cosh(c)) / (2*(cosh(c)+c*(sinh(c)-1)) * c^(n+1)), where c = A030178 = LambertW(1) = 0.56714329040978387299996866... . - _Vaclav Kotesovec_, May 21 2015

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 72*x^4/4! + 665*x^5/5! +...

%e where A(x) = 1 + x*cosh(x) + x^2*cosh(2*x) + x^3*cosh(3*x) + x^4*cosh(4*x) +...

%t CoefficientList[Series[(1-x*Cosh[x])/(1-2*x*Cosh[x]+x^2), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, May 21 2015 *)

%o (PARI) {a(n) = sum(k=0,n, n!/k! * ((n-k)^k + (-n+k)^k)/2)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = local(A=1); A = sum(m=0,n, x^m*cosh(m*x +x*O(x^n))); n!*polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = local(X=x+x*O(x^n),A=1); A = (1 - x*cosh(X)) / (1 - 2*x*cosh(X) + x^2); n!*polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A030178.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 20 2015