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 1-4 of 4 results.

A381145 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-sin(x)) ).

Original entry on oeis.org

1, 1, 3, 15, 105, 937, 10059, 124607, 1720593, 25578001, 391041299, 5628440015, 55397475705, -847789025159, -93469767131685, -5040670692970753, -236210967512228575, -10629917015586704351, -475183316832486106589, -21394016956935371375601, -975459739630268065696887
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Crossrefs

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
    a(n) = sum(k=0, n, (n+1)^(k-1)*I^(n-k)*a136630(n, k));

Formula

E.g.f. A(x) satisfies A(x) = exp( sin(x * A(x)) ).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A381148 E.g.f. A(x) satisfies A(x) = exp( sin(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 0, -11, -88, -459, -560, 27945, 502336, 5223945, 18968576, -671465123, -20909349888, -345616002627, -2437013715968, 65881260463697, 3252353828442112, 76987773739473809, 873339053357432832, -17521718791602049595, -1354633521318944473088
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Crossrefs

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
    a(n) = sum(k=0, n, (n-k+1)^(k-1)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A385424 Expansion of e.g.f. exp( -LambertW(-arcsin(x)) ).

Original entry on oeis.org

1, 1, 3, 17, 137, 1465, 19499, 311873, 5829073, 124796081, 3012319315, 80960234577, 2398138520409, 77630951407529, 2726829925494011, 103300796618253825, 4198494172961579169, 182239547736082960737, 8414068749731088539299, 411754575622058760824593
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-asin(x)))))

Formula

E.g.f. A(x) satisfies A(x) = exp( arcsin(x) * A(x) ).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A385343(n,k).
a(n) ~ n^(n-1) / (sqrt(cos(exp(-1))) * sin(exp(-1))^(n - 1/2) * exp(n - 3/2)). - Vaclav Kotesovec, Jun 28 2025

A381263 Expansion of e.g.f. exp( -LambertW(-2 * sin(x)) / 2 ).

Original entry on oeis.org

1, 1, 5, 48, 709, 14152, 356793, 10882648, 389790889, 16040853568, 745908722477, 38681745244032, 2213527304014189, 138556837227204736, 9417928265797994145, 690818806495197538816, 54391227913053881634001, 4575388875753714015748096, 409532433006878699321370197
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
    a(n) = sum(k=0, n, (2*k+1)^(k-1)*I^(n-k)*a136630(n, k));

Formula

E.g.f. A(x) satisfies A(x) = exp( sin(x) * A(x)^2 ).
a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
Showing 1-4 of 4 results.