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.

A381273 Expansion of e.g.f. exp(x * cosh(2*x)).

Original entry on oeis.org

1, 1, 1, 13, 49, 201, 2161, 12629, 102817, 1118161, 9109921, 105660765, 1223720785, 13461561881, 186666204817, 2406325357861, 33607592404033, 516511765519521, 7658010172957249, 126206019752173997, 2115466479287184241, 36218229615683409001, 666810643855970901937
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Comments

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

Crossrefs

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, 2^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^(n-k) * A185951(n,k).

A381341 Expansion of e.g.f. exp( x * cosh(sqrt(2)*x) ).

Original entry on oeis.org

1, 1, 1, 7, 25, 81, 601, 3207, 18705, 156385, 1087441, 8962823, 84001897, 732712241, 7487525865, 78537490951, 831722893217, 9804469109953, 115549730623009, 1431784628480007, 18795444460125241, 248964703826005777, 3487888859183694329, 50283005924345951111
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2025

Keywords

Comments

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

Crossrefs

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, 2^((n-k)/2)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^((n-k)/2) * A185951(n,k).

A381342 Expansion of e.g.f. exp( x * cos(sqrt(2)*x) ).

Original entry on oeis.org

1, 1, 1, -5, -23, -39, 361, 2675, 3697, -90575, -741839, 52779, 48483865, 358510985, -1225182503, -43006420829, -239523048095, 2745896185953, 54532102774753, 144304368441179, -6547928921714999, -88336890555248327, 199686588300036553, 18186115601328322515
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2025

Keywords

Comments

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

Crossrefs

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, (-2)^((n-k)/2)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} (-2)^((n-k)/2) * A185951(n,k).
Showing 1-3 of 3 results.