A194017 E.g.f.: Sum_{n>=0} x^n*cosh(n*x)^n.

1, 1, 2, 9, 120, 1745, 27600, 677257, 21712768, 744083937, 30977498880, 1600194421001, 91130331933696, 5838537271464625, 436126091418314752, 36066287271874753545, 3270603088097677639680, 332258104553585575289537, 37036273091092276854325248

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

Formula

E.g.f.: Sum_{n>=0} (x/2)^n * Sum_{k=0..n} C(n,k) * exp(n*(n-2*k)*x).

E.g.f.: 1/sqrt(1-x^2) + Sum_{n>=1} exp(-n^2*x)*[(1-sqrt(1-x^2*exp(2*n*x)))/x]^n/sqrt(1-x^2*exp(2*n*x)) + Sum_{n>=1} exp(n^2*x)*[(1-sqrt(1-x^2*exp(-2*n*x)))/x]^n/sqrt(1-x^2*exp(-2*n*x)).

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 120*x^4/4! + 1745*x^5/5! +...
where
A(x) = 1 + x*cosh(x) + x^2*cosh(2*x)^2 + x^3*cosh(3*x)^3 +...
Equivalently,
A(x) = 1 + (x/2)*(exp(x)+exp(-x)) + (x/2)^2*(exp(2*x)+exp(-2*x))^2 + (x/2)^3*(exp(3*x)+exp(-3*x))^3 +...
Also,
A(x) = 1 + (x/2)*(exp(x)+exp(-x)) + (x/2)^2*(exp(4*x)+ 2 +exp(-4*x)) + (x/2)^3*(exp(9*x)+3*exp(3*x)+3*exp(-3*x)+exp(-9*x)) + (x/2)^4*(exp(16*x)+4*exp(8*x)+ 6 +4*exp(-8*x)+exp(-16*x)) +...

Prog

(PARI) {a(n)=local(A=sum(m=0, n, x^m*cosh(m*x+x*O(x^n))^m)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(X=x+x*O(x^n), A=1/sqrt(1-X^2) + sum(m=1, n, exp(-m^2*X)/x^m/sqrt(1-X^2*exp(2*m*X))*(1-sqrt(1-x^2*exp(2*m*X)))^m) + sum(m=1, n, exp(m^2*X)/x^m/sqrt(1-x^2*exp(-2*m*X))*(1-sqrt(1-x^2*exp(-2*m*X)))^m)); n!*polcoeff(A, n)}

Crossrefs

Cf. A217017.

Sequence in context: A075538 A067965 A237999 * A135543 A372745 A316855

Adjacent sequences: A194014 A194015 A194016 * A194018 A194019 A194020

Keyword

nonn

Author

Paul D. Hanna, Aug 11 2011

Status

approved