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.

A365911 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4*k+3) / (4*k+3)! ).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 20, 1, 0, 1680, 240, 1, 369600, 102960, 4160, 168168001, 76876800, 7743840, 137225153280, 93117024001, 17091609600, 182510023324320, 172080261401600, 49615854288001, 369403226582016000, 461748751736204400, 191552892427653120
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2023

Keywords

Links

Crossrefs

Cf. A102233, A332258, A365912.
Cf. A352429, A365917.
Cf. A307978.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(sinh(x)-sin(x))/2)))

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/4)} binomial(n,4*k+3) * a(n-4*k-3).
E.g.f.: 1 / ( 1 - (sinh(x) - sin(x))/2 ).

A352428 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n,3*k+1) * a(n-3*k-1).

Original entry on oeis.org

1, 1, 2, 6, 25, 130, 810, 5881, 48806, 455706, 4727881, 53955682, 671730246, 9059714665, 131588822822, 2047796305470, 33992509701721, 599526848094850, 11195864285933682, 220692569175568729, 4579248276057441926, 99767702172338210898, 2277136869014579978473, 54336724559407913237122
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 16 2022

Keywords

Links

Crossrefs

Cf. A000670, A006154, A243664, A291973, A333881, A352429, A352430.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 3 k + 1] a[n - 3 k - 1], {k, 0, Floor[(n - 1)/3]}]; Table[a[n], {n, 0, 23}]
nmax = 23; CoefficientList[Series[1/(1 - Sum[x^(3 k + 1)/(3 k + 1)!, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\3, x^(3*k+1)/(3*k+1)!)))) \\ Seiichi Manyama, Mar 23 2022

Formula

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

A352430 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} binomial(n,5*k+1) * a(n-5*k-1).

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 721, 5054, 40488, 364896, 3654000, 40249441, 483659508, 6296246424, 88269037584, 1325861901000, 21243052172161, 361630022931666, 6518319228715302, 124018898163736536, 2483799332459535000, 52231733840672804881, 1150683180739820615582, 26502219276887376327696
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 16 2022

Keywords

Links

Crossrefs

Cf. A000670, A006154, A243666, A333882, A352428, A352429.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 5 k + 1] a[n - 5 k - 1], {k, 0, Floor[(n - 1)/5]}]; Table[a[n], {n, 0, 23}]
nmax = 23; CoefficientList[Series[1/(1 - Sum[x^(5 k + 1)/(5 k + 1)!, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\5, x^(5*k+1)/(5*k+1)!)))) \\ Seiichi Manyama, Mar 23 2022

Formula

E.g.f.: 1 / (1 - Sum_{k>=0} x^(5*k+1) / (5*k+1)!).

A365917 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4*k+5) / (4*k+5)! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 252, 0, 0, 1, 4004, 756756, 0, 1, 65756, 69837768, 11732745024, 1, 1047508, 5772957036, 3957845988096, 623360743125121, 16781260, 475191562560, 1078063276530240, 587517500395425601, 88832646060056769732, 38604505286340
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2023

Keywords

Crossrefs

Cf. A245790, A365915, A365916.
Cf. A352429, A365911.
Cf. A365898.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+x-(sinh(x)+sin(x))/2)))

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-5)/4)} binomial(n,4*k+5) * a(n-4*k-5).
E.g.f.: 1 / ( 1 + x - (sinh(x) + sin(x))/2 ).
Showing 1-4 of 4 results.

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