cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A205571 Expansion of e.g.f. 1/(1 - x*cosh(x)).

Original entry on oeis.org

1, 1, 2, 9, 48, 305, 2400, 22057, 230272, 2708001, 35412480, 509177801, 7986468864, 135718942801, 2483729876992, 48699677975145, 1018542257111040, 22634000289407297, 532557637644976128, 13226748101381102473, 345792863300174479360, 9492229607399841038961
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2012

Keywords

Comments

Radius of convergence of e.g.f. is |x| < r where r = 0.7650099545507... satisfies cosh(r) = 1/r. See A069814.

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 305*x^5/5! +...

Links

Crossrefs

Cf. A069814, A185951.
Cf. A000111, A006154, A352252.
Cf. A381206, A381207.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 13 2013 *)
  • PARI
    {a(n)=n!*polcoeff(1/(1-x*cosh(x +x*O(x^n))),n)}
  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

Formula

a(2*n-1) == 1 (mod 4), a(2*n+2) == 0 (mod 4), for n>=1.
a(n) ~ n!/(1+r*sqrt(1-r^2))*(1/r)^n, where r = A069814 = 0.7650099545507321... is the root of the equation r*cosh(r)=1. - Vaclav Kotesovec, Feb 13 2013
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
a(n) = Sum_{k=0..n} k! * A185951(n,k). - Seiichi Manyama, Feb 17 2025

A381207 Expansion of e.g.f. 1/(1 - x*cosh(x))^3.

Original entry on oeis.org

1, 3, 12, 69, 504, 4335, 43200, 490161, 6220032, 87242427, 1340305920, 22375475133, 403237638144, 7801208775399, 161245892161536, 3545854432602345, 82653484859228160, 2035605515838402291, 52814589875313573888, 1439814136866851346357, 41145786213980645621760
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A377530, A381209, A381210, A381211.
Cf. A205571, A381206.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+2)!*a185951(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * A185951(n,k).

A381377 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x * A(x)) )^2.

Original entry on oeis.org

1, 2, 6, 30, 288, 4090, 68160, 1292774, 28627200, 739821618, 21729070080, 708442911022, 25365382259712, 992297344710698, 42173572623716352, 1934344590577340790, 95175474351245230080, 5000227637170108004194, 279428527333796676894720, 16552583621200571079876158
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A185951, A381206, A381376.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = 2*sum(k=0, n, k!*binomial(2*n-k+2, k)/(2*n-k+2)*a185951(n, k));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381376.
a(n) = 2 * Sum_{k=0..n} k! * binomial(2*n-k+2,k)/(2*n-k+2) * A185951(n,k).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.