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 1-7 of 7 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

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

Cf. A350366, A351507, A351508.

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).

A384012 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^3.

Original entry on oeis.org

1, 3, 33, 630, 17247, 616770, 27264976, 1436603616, 87922855935, 6131105251425, 479931312805425, 41674568874964740, 3975727750503656820, 413360925414308633034, 46523118781014280909560, 5635356193271621706436800, 730994763063708819170060751, 101099888222006502307905386445
Offset: 0

Views

Author

Seiichi Manyama, May 17 2025

Keywords

Crossrefs

Cf. A129256, A384031, A351507, A384017, A383862.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1+k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 17}] (* Vaclav Kotesovec, May 18 2025 *)
  • PARI
    a(n) = sum(i=0, n, sum(j=0, 2*n-i, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, 2*n-i-j+1, 1))));

Formula

a(n) = Sum_{0<=i, j, k<=n and i+j+k=2*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^(n - 1/2) / (2^(2*n + 5/2) * sqrt(Pi*(w-1)) * exp(n) * (3*w-2)^n), where w = -LambertW(-1,-2*exp(-2/3)/3) = 1.4293552275170056487105688431034768889546376014196... - Vaclav Kotesovec, May 18 2025

A351508 a(n) = [x^n] Product_{k=1..n} 1/(1 - k*x)^n.

Original entry on oeis.org

1, 1, 23, 1386, 162154, 31354800, 9078595483, 3682549444112, 1994756395887972, 1391788744738729470, 1216130179327397765925, 1301126343608005909401330, 1673298722590019165433540916, 2547164111922284803722749855516
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2022

Keywords

Crossrefs

Cf. A007820, A350376, A383862, A384060.
Cf. A351507, A384060.

Programs

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

Formula

a(n) ~ exp(n + 5/3) * n^(2*n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, Feb 18 2022
a(n) = Sum_{x_1, x_2,..., x_n >= 0 and x_1 + x_2 + ... + x_n = n} Product_{k=1..n} Stirling2(x_k + n,n). - Seiichi Manyama, May 18 2025

A384017 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^5.

Original entry on oeis.org

1, 5, 100, 3510, 177370, 11732175, 960453825, 93791830160, 10644367637490, 1376936603007075, 200002385378370350, 32233130183113838550, 5708169533474858008905, 1101836121788665346133960, 230256048227047074266497400, 51791322674249971562728368000
Offset: 0

Views

Author

Seiichi Manyama, May 18 2025

Keywords

Comments

From Vaclav Kotesovec, May 19 2025: (Start)
In general, for m > 1, [x^n] Product_{k=0..n} (1 + k*x)^m ~ m^(m*(n + 1/2)) * w^(m*n + (m+1)/2) * n^(n - 1/2) / (sqrt(2*Pi*(w-1)) * exp(n) * (m-1)^((m-1)*n + (m+1)/2) * (m*w-m+1)^n), where w = -LambertW(-1,-(m-1)*exp(-(m-1)/m)/m).
The general formula is valid even for m=n, where after modifications we get the formula for A351507. (End)

Crossrefs

Cf. A000142 (m=1), A129256 (m=2), A384012 (m=3), A384031 (m=4), A351507 (m=n).

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1+k*x)^5, {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 19 2025 *)
  • PARI
    a(n) = polcoef(prod(k=1, n, 1+k*x)^5, n);

Formula

a(n) = Sum_{0<=i, j, k, l, m<=n and i+j+k+l+m=4*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1) * Stirling1(n+1,m+1)|.
a(n) ~ 5^(5*n + 5/2) * w^(5*n+3) * n^(n - 1/2) / (2^(8*n + 13/2) * sqrt(Pi*(w-1)) * exp(n) * (5*w-4)^n), where w = -LambertW(-1,-4*exp(-4/5)/5) = 1.2308422097842590367678406745433500325966... - Vaclav Kotesovec, May 19 2025

A384091 a(n) = [x^n] Product_{k=1..n} (1 + k^2*x)^n.

Original entry on oeis.org

1, 1, 33, 6968, 4503078, 6507545775, 17683339661956, 80849884332530600, 575530003415681613468, 6023356562522188931288775, 88682105895482127774508529242, 1773600518272635675832361778156960, 46830898160739235037404595987069052560, 1594447058825655577475889095097916983404652
Offset: 0

Views

Author

Vaclav Kotesovec, May 19 2025

Keywords

Crossrefs

Cf. A351507, A001044, A384092.

Programs

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

Formula

a(n) ~ exp(n + 3/5) * n^(3*n - 1/2) / (sqrt(2*Pi) * 3^n).

A384089 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^n.

Original entry on oeis.org

1, 0, 1, 63, 7206, 1357300, 384271700, 153027592116, 81648987014364, 56259916067074896, 48646018448463951450, 51584263505394472459750, 65833976467770842558152992, 99553004175105699906002335098, 176031670802373999913671973955080, 359870756416991348769957239299854000
Offset: 0

Views

Author

Seiichi Manyama, May 19 2025

Keywords

Crossrefs

Cf. A342111, A384018, A384029.
Cf. A351507.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + k*x)^n, {k, 0, n-1}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2025 *)
  • PARI
    a(n) = polcoef(prod(k=0, n-1, 1+k*x)^n, n);

Formula

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,x_k)|.
a(n) ~ exp(n - 5/3) * n^(2*n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2025
Showing 1-7 of 7 results.

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