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.

A306347 Expansion of e.g.f. exp((sin(x) + sinh(x))/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 128, 389, 1904, 9329, 38040, 132147, 542648, 3283633, 20997824, 114657097, 536178880, 2784500161, 19876061312, 153326461311, 1034551839872, 6051063485481, 38079448046208, 312420426154893, 2785055242928768, 22141255520251313
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2019

Keywords

Comments

Number of partitions of n-set into blocks congruent to 1 mod 4.

Links

Crossrefs

Cf. A002017, A003724, A005046, A013025, A016813, A035451, A291975, A307978, A307979, A351969.

Programs

  • Mathematica
    nmax = 28; CoefficientList[Series[Exp[(Sin[x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 28}]
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sin(x)+sinh(x))/2))) \\ Seiichi Manyama, Mar 17 2022
  • PARI
    a(n) = if(n==0, 1, sum(k=0, (n-1)\4, binomial(n-1, 4*k)*a(n-4*k-1))); \\ Seiichi Manyama, Mar 17 2022

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * a(n-4*k-1). - Seiichi Manyama, Mar 17 2022

A351968 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(k) * a(n-3*k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 37, 114, 478, 1907, 6777, 28414, 148580, 758930, 3580294, 18982050, 117888762, 720679726, 4193516446, 26798335830, 191775198574, 1353198262531, 9303932353127, 69303156652024, 559295471922890, 4454686099742810, 35198016469190740
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 26 2022

Keywords

Crossrefs

Cf. A138314, A351969.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 3 k] a[k] a[n - 3 k - 1], {k, 0, Floor[(n - 1)/3]}]; Table[a[n], {n, 0, 27}]

Formula

E.g.f.: exp( Sum_{n>=0} a(n) * x^(3*n+1) / (3*n+1)! ).

A352437 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4*k+1) * a(k) * a(n-4*k-1).

Original entry on oeis.org

1, 1, 2, 6, 24, 121, 732, 5166, 41664, 378002, 3810532, 42254256, 511145184, 6698543286, 94536850576, 1429503114816, 23056734645504, 395129481418584, 7169760547544240, 137325429039660432, 2768685793537965504, 58611883690279194361, 1299873446056852916876
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 16 2022

Keywords

Crossrefs

Cf. A351969, A352435, A352436.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 4 k + 1] a[k] a[n - 4 k - 1], {k, 0, Floor[(n - 1)/4]}]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: 1 / (1 - Sum_{n>=0} a(n) * x^(4*n+1) / (4*n+1)!).
Showing 1-3 of 3 results.

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

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