A384017 -id:A384017 - cp's OEIS frontend

A129256 Central coefficient of Product_{k=0..n} (1+k*x)^2.

1, 2, 13, 144, 2273, 46710, 1184153, 35733376, 1251320145, 49893169050, 2232012515445, 110722046632560, 6032418472347265, 358103844593876654, 23007314730623658225, 1590611390957425536000, 117745011140615270168865

Offset: 0

Paul D. Hanna, Apr 06 2007

nonn

			This sequence equals the central terms of the triangle in which the g.f. of row n is (1+x)^2*(1+2x)^2*(1+3x)^2*...*(1+n*x)^2, as illustrated by:
  (1);
   1, (2),  1;
   1,  6, (13),  12,     4;
   1, 12,  58, (144),  193,    132,      36;
   1, 20, 170,  800, (2273),  3980,    4180,   2400,    576;
   1, 30, 395, 3000, 14523, (46710), 100805, 143700, 129076, 65760, 14400;
  ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..354

Cf. A008275 (Stirling1 numbers), A187235, A238261, A246117, A254882, A350376.

Cf. A384012, A384031, A384017.

Flatten[{1,Table[Coefficient[Expand[Product[(1+k*x),{k,0,n}]^2],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

PARI
```
a(n)=polcoeff(prod(k=0,n,1+k*x)^2,n)
```

{a(n)=(-1)^n*sum(k=0,n,stirling(n+1,k+1,1)*stirling(n+1,n-k+1,1))} \\ Paul D. Hanna, Jul 16 2009

a(n) = (-1)^n*Sum_{k=0..n} Stirling1(n+1,k+1)*Stirling1(n+1,n-k+1). - Paul D. Hanna, Jul 16 2009

a(n) ~ c * d^n * (n-1)!, where d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.910814964568255..., c = (-LambertW(-1, -exp(-1/2)/2))^(3/2)/(sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.851946112888790982829578047527831525434714038256... . - Vaclav Kotesovec, Feb 10 2015, updated May 14 2025

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

Seiichi Manyama, May 17 2025

nonn

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

Table[SeriesCoefficient[Product[(1+k*x)^4, {k, 1, n}], {x, 0, n}], {n, 0, 16}] (* Vaclav Kotesovec, May 18 2025 *)

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)))));

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

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

Seiichi Manyama, Feb 12 2022

nonn

Cf. A000142, A129256, A384012, A384031.

Cf. A351508, A384017.

Table[SeriesCoefficient[Product[(1 + k*x)^n, {k,1,n}], {x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 18 2022 *)

a(n) = polcoef(prod(k=1, n, 1+k*x)^n, n);

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

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

Seiichi Manyama, May 17 2025

nonn

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

Table[SeriesCoefficient[Product[(1+k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 17}] (* Vaclav Kotesovec, May 18 2025 *)

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))));

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

cp's OEIS Frontend

A129256 Central coefficient of Product_{k=0..n} (1+k*x)^2.

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

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

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

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