A384012 -id:A384012 - 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

A383862 a(n) = [x^n] Product_{k=0..n} 1/(1 - k*x)^3.

Original entry on oeis.org

1, 3, 48, 1386, 58278, 3225915, 221726711, 18216234288, 1741626159966, 189977753488050, 23285057201978520, 3168272346322892094, 473878954663846060735, 77281168674525142984020, 13647787698908399220563400, 2594721838238358445753776000, 528401900314147344955336365822

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A007820, A350376, A384022, A384060.

Cf. A384012.

Table[SeriesCoefficient[Product[1/(1-k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 16}] (* Vaclav Kotesovec, May 18 2025 *)
(* or *)
Table[Sum[StirlingS2[i + n, n] * StirlingS2[j + n, n] * StirlingS2[2*n - i - j, n], {i, 0, n}, {j, 0, n-i}], {n, 0, 16}] (* Vaclav Kotesovec, May 18 2025 *)

a(n) = sum(i=0, n, sum(j=0, n-i, stirling(i+n, n, 2)*stirling(j+n, n, 2)*stirling(2*n-i-j, n, 2)));

a(n) = Sum_{i, j, k>=0 and i+j+k=n} Stirling2(i+n,n) * Stirling2(j+n,n) * Stirling2(k+n,n).

a(n) ~ 2^(8*n + 3/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * exp(n) * 3^(3*n + 3/2) * (4 - 3*w)^n * w^(3*n + 1)), where w = -LambertW(-4*exp(-4/3)/3) = 0.727473355414332993149219573314579663... - 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

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

Seiichi Manyama, May 18 2025

nonn

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)

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

Table[SeriesCoefficient[Product[(1+k*x)^5, {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 19 2025 *)

a(n) = polcoef(prod(k=1, n, 1+k*x)^5, n);

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

A384087 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^3.

Original entry on oeis.org

1, 6, 162, 7848, 552000, 51035310, 5853933666, 802178739936, 127879052859648, 23252775004089990, 4750089647035004250, 1077069265550569663416, 268437124701985949614944, 72940650531961450912140558, 21461129870889481564510048050, 6797577340761206051865208521600, 2306127185536355501260494657447936

Offset: 0

Vaclav Kotesovec, May 19 2025

nonn

Cf. A384012, A383862.

Cf. A350366, A384086, A384088, A351764.

Table[SeriesCoefficient[Product[(1+k*x)^3/(1-k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 16}]

a(n) ~ c * d^n * n! / n, where d = 22.56625698335414867480351407039325848948214595770919713967057... and c = 0.403760467212667768540403611728406212428403946576093482938996...

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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A383862 a(n) = [x^n] Product_{k=0..n} 1/(1 - k*x)^3.

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

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

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A384087 a(n) = [x^n] Product_{k=1..n} ((1 + k*x)/(1 - k*x))^3.

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Formula

A384087 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^3.