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.

A357902 a(n) = Sum_{k=0..floor(n/4)} |Stirling1(n - 3*k,k)|.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 6, 25, 123, 731, 5090, 40595, 364650, 3641903, 40026609, 480029801, 6237662582, 87298953249, 1309161984315, 20942605407386, 355971044728635, 6406714801013007, 121715861296354116, 2434125806029297550, 51113325326999860554, 1124432395936987325868
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2022

Keywords

Crossrefs

Cf. A000142, A343579, A357901.

Programs

  • PARI
    a(n) = sum(k=0, n\4, abs(stirling(n-3*k, k, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+x^3)))

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (j + x^3).

A357919 a(n) = Sum_{k=0..floor(n/3)} Stirling1(n - 2*k,k).

Original entry on oeis.org

1, 0, 0, 1, -1, 2, -5, 21, -109, 671, -4772, 38591, -350036, 3520830, -38903271, 468490350, -6107642906, 85704534787, -1288021805215, 20641247413120, -351374756822383, 6332030169529731, -120427840368046909, 2410627702030000447, -50661193580285096086
Offset: 0

Views

Author

Seiichi Manyama, Oct 20 2022

Keywords

Crossrefs

Cf. A357901, A357920.

Programs

  • Maple
    A357919 := proc(n)
    add(stirling1(n-2*k,k),k=0..n/3) ;
end proc:
seq(A357919(n),n=0..70) ; # R. J. Mathar, Mar 13 2023
  • PARI
    a(n) = sum(k=0, n\3, stirling(n-2*k, k, 1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k*prod(j=0, k-1, j-x^2)))

Formula

G.f.: Sum_{k>=0} (-x)^k * Product_{j=0..k-1} (j - x^2).

A357931 a(n) = Sum_{k=0..floor(n/3)} |Stirling1(n - 2*k,n - 3*k)|.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 7, 13, 27, 57, 120, 262, 593, 1361, 3171, 7559, 18356, 45186, 112927, 286689, 737641, 1921639, 5070154, 13540352, 36566737, 99830013, 275459693, 767798853, 2160953618, 6139721116, 17604534427, 50924095081, 148570523479, 437071675997
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2022

Keywords

Crossrefs

Cf. A124380, A357932, A357933.
Cf. A353223, A357901, A357925.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n-2k,n-3k]],{k,0,Floor[n/3]}],{n,0,40}] (* Harvey P. Dale, Nov 01 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, abs(stirling(n-2*k, n-3*k, 1)));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, 1+j*x^2)))

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (1 + j * x^2).

A357922 a(n) = Sum_{k=0..floor(n/5)} |Stirling1(n - 4*k,k)|.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 6, 24, 121, 723, 5051, 40370, 363154, 3630565, 39929874, 479111219, 6228047601, 87188921464, 1307794924973, 20924276449014, 355707232027825, 6402657184129671, 121649439722758345, 2432972744390660437, 51092165603897459951
Offset: 0

Views

Author

Seiichi Manyama, Oct 20 2022

Keywords

Crossrefs

Cf. A000142, A343579, A357901, A357902.
Cf. A357920.

Programs

  • PARI
    a(n) = sum(k=0, n\5, abs(stirling(n-4*k, k, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+x^4)))

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (j + x^4).
Showing 1-4 of 4 results.

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