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.

A356362 a(n) = Sum_{k=0..floor(n/3)} n^k * Stirling1(n,3*k).

Original entry on oeis.org

1, 0, 0, 3, -24, 175, -1314, 10339, -84448, 696429, -5444700, 32897601, 53444304, -8071238721, 235927045536, -5630771421765, 126525509087232, -2799633511755963, 62154971516786616, -1396560425289392007, 31880150704745078400, -740188445913015688953
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2022

Keywords

Links

Crossrefs

Cf. A356361, A356363.
Cf. A357834.

Programs

  • PARI
    a(n) = sum(k=0, n\3, n^k*stirling(n, 3*k, 1));
  • PARI
    a(n) = n!*polcoef(sum(k=0, n\3, n^k*log(1+x+x*O(x^n))^(3*k)/(3*k)!), n);
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(v=n^(1/3), w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-v, n)+Pochhammer(-v*w, n)+Pochhammer(-v*w^2, n))/3;

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . a(n) = n! * [x^n] F(n^(1/3) * log(1+x)).
a(n) = (-1)^n * ( (-n^(1/3))_n + (-n^(1/3)*w)_n + (-n^(1/3)*w^2)_n )/3, where (x)_n is the Pochhammer symbol.

A357835 a(n) = Sum_{k=0..floor((n-1)/3)} Stirling1(n,3*k+1).

Original entry on oeis.org

0, 1, -1, 2, -5, 14, -35, -14, 1701, -26418, 351351, -4622982, 62705643, -890078826, 13297263525, -209438953542, 3477446002485, -60803484275898, 1117975706702127, -21580455768575886, 436591651807054107, -9241512424454751714, 204338436416329792941
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2022

Keywords

Links

Crossrefs

Cf. A357834, A357836.

Programs

  • PARI
    a(n) = sum(k=0, (n-1)\3, stirling(n, 3*k+1, 1));
  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N\3, log(1+x)^(3*k+1)/(3*k+1)!))))
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-1, n)+w^2*Pochhammer(-w, n)+w*Pochhammer(-w^2, n))/3;

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(log(1+x)).
a(n) = (-1)^n * ( (-1)_n + w^2 * (-w)_n + w * (-w^2)_n )/3, where (x)_n is the Pochhammer symbol.

A357836 a(n) = Sum_{k=0..floor((n-2)/3)} Stirling1(n,3*k+2).

Original entry on oeis.org

0, 0, 1, -3, 11, -49, 259, -1589, 11109, -87171, 758121, -7229859, 74905467, -836159961, 9980000667, -126422745813, 1686902233653, -23512989735963, 338917341235473, -4982536435536387, 73087736506615467, -1025163078325255233, 12286912220375608179
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2022

Keywords

Links

Crossrefs

Cf. A357834, A357835.

Programs

  • PARI
    a(n) = sum(k=0, (n-2)\3, stirling(n, 3*k+2, 1));
  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, log(1+x)^(3*k+2)/(3*k+2)!))))
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-1, n)+w*Pochhammer(-w, n)+w^2*Pochhammer(-w^2, n))/3;

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(log(1+x)).
a(n) = (-1)^n * ( (-1)_n + w * (-w)_n + w^2 * (-w^2)_n )/3, where (x)_n is the Pochhammer symbol.
Showing 1-3 of 3 results.

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