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.

A352638 Expansion of e.g.f. 1/(1 - 3*sin(x)).

Original entry on oeis.org

1, 3, 18, 159, 1872, 27543, 486288, 10016619, 235798272, 6244714443, 183756215808, 5947907121879, 210026879004672, 8034293365747743, 330982609573398528, 14609181655918083939, 687820834029346947072, 34407546247054875367443, 1822450167175258689896448
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Cf. A000111, A007289.
Cf. A107403, A352640.
Cf. A136630.

Programs

  • Mathematica
    With[{m = 17}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*Sin[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*sin(x))))
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / (2^(3/2) * arcsin(1/3)^(n+1)). - Vaclav Kotesovec, Mar 26 2022
a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

A381278 Expansion of e.g.f. exp(sin(3*x) / 3).

Original entry on oeis.org

1, 1, 1, -8, -35, -8, 1117, 6328, -19943, -513728, -2096711, 30574720, 447401845, 23791744, -59033858219, -527680180736, 4971322421425, 144677554315264, 430091284739185, -27641200139694080, -398305237630617971, 2876369985206861824, 145441158283475935309
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Cf. A002017, A009210.
Cf. A136630, A352640, A381286.

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, (3*I)^(n-k)*a136630(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-9)^k * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} (3*i)^(n-k) * A136630(n,k), where i is the imaginary unit.

A352639 Expansion of e.g.f. exp(2*sin(x)).

Original entry on oeis.org

1, 2, 4, 6, 0, -46, -192, -266, 1792, 14114, 34816, -171930, -2027520, -6522382, 34750464, 496296022, 1748500480, -12731696062, -186550845440, -617309234490, 7292215885824, 99199654760978, 248883934396416, -5836506132182090, -69729013345550336
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Cf. A002017, A136630, A352640.
Cf. A007289, A352279, A352642.

Programs

  • Mathematica
    With[{m = 24}, Range[0, m]! * CoefficientList[Series[Exp[2*Sin[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sin(x))))
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*binomial(n-1, 2*k)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^k * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025
Showing 1-3 of 3 results.

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