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.

A351507 a(n) = [x^n] Product_{k=1..n} (1 + k*x)^n.

Original entry on oeis.org

1, 1, 13, 630, 65446, 11732175, 3222746276, 1257489289280, 661150527657180, 450529025613124095, 386183537239831846450, 406654820487854059966416, 516014868401631381045209376, 776565429016998902169538130936, 1367544537916924083498367095477800
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2022

Keywords

Crossrefs

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + k*x)^n, {k,1,n}], {x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 18 2022 *)
  • PARI
    a(n) = polcoef(prod(k=1, n, 1+k*x)^n, n);

Formula

a(n) ~ exp(n + 1/3) * n^(2*n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, Feb 18 2022
a(n) = Sum_{0 <= x_1, x_2,..., x_n <= n and x_1 + x_2 + ... + x_n = (n-1)*n} Product_{k=1..n} |Stirling1(n + 1,x_k + 1)|. - Seiichi Manyama, May 18 2025

A351764 a(n) = [x^n] Product_{k=1..n} (1 + k*x)^n / (1 - k*x)^n.

Original entry on oeis.org

1, 2, 72, 7848, 1728000, 641258850, 360403076376, 285818177146208, 304172586944446464, 418400927094822149970, 722587619114932445325000, 1530927286486636135080191736, 3904621941927926455303092180480, 11801667653769333351640692783069714
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 18 2022

Keywords

Crossrefs

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + k*x)^n / (1 - k*x)^n, {k,1,n}], {x,0,n}], {n,0,20}]
  • PARI
    a(n) = my(x='x+O(x^(n+1))); polcoef(prod(k=1, n, (1+k*x)^n/ (1-k*x)^n), n); \\ Michel Marcus, Feb 19 2022

Formula

a(n) ~ exp(n + 1) * n^(2*n - 1/2) / sqrt(2*Pi).

A384060 a(n) = [x^n] Product_{k=0..n} 1/(1 - k*x)^4.

Original entry on oeis.org

1, 4, 82, 3024, 162154, 11438280, 1001454024, 104777127616, 12755141675754, 1771354690734420, 276386332002204450, 47870892086756660064, 9113932961179205496744, 1891845220489637114281216, 425240943851497448491619600, 102899751348092720847554016000
Offset: 0

Views

Author

Vaclav Kotesovec, May 18 2025

Keywords

Comments

In general, for m>=1, [x^n] Product_{k=0..n} 1/(1 - k*x)^m ~ (m+1)^((m+1)*n + (m-1)/2) * n^(n - 1/2) / (sqrt(2*Pi*(1-w)) * exp(n) * (m+1-m*w)^n * m^(m*(n + 1/2)) * w^(m*n + (m-1)/2)), where w = -LambertW(-(m+1)*exp(-(m+1)/m)/m).
The general formula is valid even for m=n, where after modifications we get the formula for A351508.

Crossrefs

Cf. A007820 (m=1), A350376 (m=2), A383862 (m=3), A351508 (m=n).
Cf. A384031.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1-k*x)^4, {k, 1, n}], {x, 0, n}], {n, 0, 15}]

Formula

a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n} Stirling2(i+n,n) * Stirling2(j+n,n) * Stirling2(k+n,n) * Stirling2(l+n,n). - Seiichi Manyama, May 18 2025
a(n) ~ 5^(5*n + 3/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * 2^(8*n + 9/2) * exp(n) * (5 - 4*w)^n * w^(4*n + 3/2)), where w = -LambertW(-5*exp(-5/4)/4) = 0.7857872456211833502961937693700363613539172187... - Vaclav Kotesovec, May 18 2025

A384092 a(n) = [x^n] Product_{k=1..n} 1/(1 - k^2*x)^n.

Original entry on oeis.org

1, 1, 67, 19316, 14842986, 23959995900, 70300141076691, 340026368533209120, 2526875675012579004324, 27358621384723375076245950, 414013875603209906596527455633, 8469874364125222067804767445806552, 227937197746419681734617268030982470980, 7887251806534473871432104574423885714752540
Offset: 0

Views

Author

Vaclav Kotesovec, May 19 2025

Keywords

Crossrefs

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1-k^2*x)^n, {k, 0, n}], {x, 0, n}], {n, 0, 15}]

Formula

a(n) ~ exp(n + 12/5) * n^(3*n - 1/2) / (sqrt(2*Pi) * 3^n).
Showing 1-4 of 4 results.