A384031 -id:A384031 - 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

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

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

Vaclav Kotesovec, May 18 2025

nonn

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.

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

Cf. A384031.

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

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

A384029 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^4.

Original entry on oeis.org

1, 0, 6, 180, 7206, 370880, 23477380, 1768061064, 154544373158, 15387101825184, 1719596420272980, 213181689525888600, 29036623040055512332, 4310582688852993653568, 692756995680614782818992, 119830419866883597939018000, 22198322332579642585088580870, 4384714751330840129324051474880

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A342111, A384018.
Cf. A384027, A384030.
Cf. A384031.

a(n) = sum(i=0, n, sum(j=0, 3*n-i, sum(k=0, 3*n-i-j, abs(stirling(n, i, 1)*stirling(n, j, 1)*stirling(n, k, 1)*stirling(n, 3*n-i-j-k, 1)))));

a(n) = Sum_{0<=i, j, k, l<=n and i+j+k+l=3*n} |Stirling1(n,i) * Stirling1(n,j) * Stirling1(n,k) * Stirling1(n,l)|.

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

A382925 a(n) = [x^(3n)] Product_{k=0..n} (1 + kx)^4.

Original entry on oeis.org

1, 4, 248, 61320, 39194896, 51699564000, 122482878310656, 474300956527856640, 2804126507444905046272, 24036712401508315774848000, 286889291626307627568309995520, 4615084616716397442547883972818944, 97421519516367186622078306709619806208

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A129256, A384025.

Cf. A384031, A384032.

Abs[Table[Sum[StirlingS1[n+1, i+1] * StirlingS1[n+1, j+1] * StirlingS1[n+1, k+1] * StirlingS1[n+1, n-i-j-k+1], {i, 0, n}, {j, 0, n-i}, {k, 0, n-i-j}], {n, 0, 15}]] (* Vaclav Kotesovec, May 22 2025 *)

a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, 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, n-i-j-k+1, 1)))));

a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=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^(3*n - 1/2) / (sqrt(Pi*(w-1)) * exp(3*n) * (4*w-1)^(3*n)), where w = -LambertW(-1, -exp(-1/4)/4) = 2.5866629822630538811828... - Vaclav Kotesovec, May 22 2025

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A382925, A384031.

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 *)

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)))));

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

A384088 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^4.

Original entry on oeis.org

1, 8, 288, 18528, 1728000, 211687080, 32159822688, 5835397918336, 1231573968949248, 296447550279133320, 80158746419240852000, 24057027574081163030688, 7935414295799696292767232, 2853706409310576479751168168, 1111199574070700473937862463200, 465782420445680979210397280524800

Offset: 0

Vaclav Kotesovec, May 19 2025

nonn

Cf. A384031, A384060.

Cf. A350366, A384086, A384087, A351764.

Table[SeriesCoefficient[Product[(1+k*x)^4/(1-k*x)^4, {k, 1, n}], {x, 0, n}], {n, 0, 16}]

a(n) ~ c * d^n * n! / n, where d = 29.85915450232266280267400661836716424701025678171993103713550551... and c = 0.415660498916272367812330643610916948922178337726778287649763513...

cp's OEIS Frontend

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

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

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A384029 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^4.

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

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A382925 a(n) = [x^(3*n)] Product_{k=0..n} (1 + k*x)^4.

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

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A384088 a(n) = [x^n] Product_{k=1..n} ((1 + k*x)/(1 - k*x))^4.

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A382925 a(n) = [x^(3n)] Product_{k=0..n} (1 + kx)^4.

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

A384088 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^4.