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.

A381144 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * cos(x)) ).

Original entry on oeis.org

1, 1, 3, 13, 65, 221, -2933, -120903, -3104127, -71637191, -1562635789, -31373685947, -505087300991, -1692007785259, 402032879446395, 28152810613025521, 1423083552938781697, 62552808878706976625, 2459148829654813484131, 82692880516086149155581
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 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, (n+1)^(k-1)*I^(n-k)*a185951(n, k));

Formula

E.g.f. A(x) satisfies A(x) = exp( x * A(x) * cos(x * A(x)) ).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381146 E.g.f. A(x) satisfies A(x) = exp( x * cos(x * A(x)) ).

Original entry on oeis.org

1, 1, 1, -2, -35, -264, -1019, 9864, 302905, 4181824, 23080201, -632195200, -25239729899, -478790195584, -2682065360883, 188875659540736, 8843706554450033, 203538869514047488, 751681101659548177, -169782541027003551744, -8866196526809624969139
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 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, (n-k+1)^(k-1)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381261 Expansion of e.g.f. exp( -LambertW(-2 * x * cos(x)) / 2 ).

Original entry on oeis.org

1, 1, 5, 46, 669, 13176, 328153, 9889328, 349998169, 14232282112, 653960139021, 33511444515968, 1894938691013173, 117209395966704640, 7872535432641217185, 570622024676568564736, 44395462114163659522353, 3690312836780077587120128, 326399124496126009678138261
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*k+1)^(k-1)*I^(n-k)*a185951(n, k));

Formula

E.g.f. A(x) satisfies A(x) = exp( x * cos(x) * A(x)^2 ).
a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.
Showing 1-3 of 3 results.