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.

A384031 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^4.

Original entry on oeis.org

1, 4, 62, 1680, 65446, 3334800, 210218956, 15803243456, 1380404187558, 137419388080920, 15359405910256580, 1904647527097204032, 259511601503239509004, 38539384808775589973416, 6195988524478342471690200, 1072149116496356641327200000, 198683315255720972000976370950
Offset: 0

Views

Author

Seiichi Manyama, May 17 2025

Keywords

Crossrefs

Cf. A129256, A384012, A384017.
Cf. A384060.
Cf. A382925, A384032, A384029, A351507.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1+k*x)^4, {k, 1, n}], {x, 0, n}], {n, 0, 16}] (* Vaclav Kotesovec, May 18 2025 *)
  • PARI
    a(n) = sum(i=0, n, sum(j=0, 3*n-i, sum(k=0, 3*n-i-j, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, k+1, 1)*stirling(n+1, 3*n-i-j-k+1, 1)))));

Formula

a(n) = Sum_{0<=i, j, k, l<=n and i+j+k+l=3*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1)|.
a(n) ~ 2^(8*n + 7/2) * w^(4*n + 5/2) * n^(n - 1/2) / (sqrt(Pi*(w-1)) * 3^(3*n + 5/2) * exp(n) * (4*w-3)^n), where w = -LambertW(-1,-3*exp(-3/4)/4) = 1.300200741659068588153265179374583756429... - Vaclav Kotesovec, May 18 2025

A384032 a(n) = [x^(2*n)] Product_{k=0..n} (1 + k*x)^4.

Original entry on oeis.org

1, 6, 321, 46364, 13052881, 6077950570, 4237586784577, 4137911590389080, 5394217192300621089, 9055251708372687577550, 19032397641903957029149569, 48970167155426122072661229684, 151429299992138418402024853511537, 554184682895238619253412365302575346
Offset: 0

Views

Author

Seiichi Manyama, May 17 2025

Keywords

Crossrefs

Cf. A382925, A384031.

Programs

  • Mathematica
    Table[Sum[StirlingS1[n+1, i+1] * StirlingS1[n+1, j+1] * StirlingS1[n+1, k+1] * StirlingS1[n+1, 2*n-i-j-k+1], {i, 0, n}, {j, 0, 2*n-i}, {k, 0, 2*n-i-j}], {n, 0, 15}] (* Vaclav Kotesovec, May 22 2025 *)
  • PARI
    a(n) = sum(i=0, n, sum(j=0, 2*n-i, sum(k=0, 2*n-i-j, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, k+1, 1)*stirling(n+1, 2*n-i-j-k+1, 1)))));

Formula

a(n) = Sum_{0<=i, j, k, l<=n and i+j+k+l=2*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1)|.
a(n) ~ 2^(4*n + 1) * w^(4*n + 5/2) * n^(2*n - 1/2) / (sqrt(Pi*(w-1)) * exp(2*n) * (2*w-1)^(2*n)), where w = -LambertW(-1, -exp(-1/2)/2) = 1.7564312086261696769827376166... - Vaclav Kotesovec, May 22 2025

A384025 a(n) = [x^(2*n)] Product_{k=0..n} (1 + k*x)^3.

Original entry on oeis.org

1, 3, 66, 3815, 424428, 77530530, 21106440064, 8021533034676, 4060456997959152, 2642189599046492000, 2149789283054191431744, 2139041823964877704864992, 2555760236856152336740829440, 3611539707805518014521602175296, 5958533262158042791156143146398464
Offset: 0

Views

Author

Seiichi Manyama, May 17 2025

Keywords

Crossrefs

Cf. A129256, A382925.

Programs

  • Mathematica
    Table[Sum[StirlingS1[n+1, i+1] * StirlingS1[n+1, j+1] * StirlingS1[n+1, n-i-j+1], {i, 0, n}, {j, 0, n-i}], {n, 0, 15}] (* Vaclav Kotesovec, May 22 2025 *)
  • PARI
    a(n) = sum(i=0, n, sum(j=0, n-i, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, n-i-j+1, 1))));

Formula

a(n) = Sum_{i, j, k>=0 and i+j+k=n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1)|.
a(n) ~ 3^(3*n + 3/2) * w^(3*n+2) * n^(2*n - 1/2) / (sqrt(2*Pi*(w-1)) * exp(2*n) * (3*w-1)^(2*n)), where w = -LambertW(-1, -exp(-1/3)/3) = 2.23714702777371681804347369... - Vaclav Kotesovec, May 22 2025
Showing 1-3 of 3 results.

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