A350376 -id:A350376 - cp's OEIS frontend

A007820 Stirling numbers of second kind S(2n,n).

1, 1, 7, 90, 1701, 42525, 1323652, 49329280, 2141764053, 106175395755, 5917584964655, 366282500870286, 24930204590758260, 1850568574253550060, 148782988064375309400, 12879868072770626040000, 1194461517469807833782085, 118144018577011378596484455

Offset: 0

kemp(AT)sads.informatik.uni-frankfurt.de (Rainer Kemp)

Chan and Manna prove that a(n) is odd if and only if n is in A003714. - Jason Kimberley, Sep 14 2009
The number of ways to partition a set of 2*n elements into n disjoint subsets. - Vladimir Reshetnikov, Oct 10 2016
Conjecture: a(2*n+1) is divisible by (2*n + 1)^2. - Peter Bala, Mar 30 2025

			G.f.: A(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 +...,
where A(x) = 1 + 1^2*x*exp(-1*x) + 2^4*exp(-2^2*x)*x^2/2! + 3^6*exp(-3^2*x)*x^3/3! + 4^8*exp(-4^2*x)*x^4/4! + 5^10*exp(-5^2*x)*x^5/5! + ... - _Paul D. Hanna_, Oct 17 2012

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

Alois P. Heinz, Table of n, a(n) for n = 0..345 (terms n = 1..200 from Vincenzo Librandi)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O-Y. Chan and D. V. Manna, Divisibility properties of Stirling numbers of the second kind [From _Jason Kimberley_, Sep 14 2009]

Cf. A002465, A008277, A187646, A191236, A217913, A217914, A217915, A247238.

Cf. A350376, A383862, A384060.

A007820 := proc(n) Stirling2(2*n,n) ; end proc:
seq(A007820(n),n=0..20) ; # R. J. Mathar, Mar 15 2011

Table[StirlingS2[2n, n], {n, 1, 12}] (* Emanuele Munarini, Mar 12 2011 *)

makelist(stirling2(2*n,n),n,0,12); /* Emanuele Munarini, Mar 12 2011 */

a(n)=stirling(2*n,n,2); /* Joerg Arndt, Jul 01 2011 */

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), n)} \\ Paul D. Hanna, Oct 17 2012

{a(n)=polcoeff(sum(m=1,n,(m^2)^m*exp(-m^2*x+x*O(x^n))*x^m/m!),n)} \\ Paul D. Hanna, Oct 17 2012

from sympy.functions.combinatorial.numbers import stirling
def A007820(n): return stirling(n<<1,n) # Chai Wah Wu, Jun 09 2025

[stirling_number2(2*i,i) for i in range(1,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = A048993(2n,n). - R. J. Mathar, Mar 15 2011
Asymptotic: a(n) ~ (4*n/(e*z*(2-z)))^n/sqrt(2*Pi*n*(z-1)), where z = A256500 = 1.59362426... is a root of the equation exp(z)*(2-z)=2. - Vaclav Kotesovec, May 30 2011
a(n) = 1/n! * Sum_{k = 0..n} binomial(n,k)*(-1)^k*(n-k)^(2*n). - Emanuele Munarini, Jul 01 2011
a(n) = [x^n] 1 / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Oct 17 2012
O.g.f.: Sum_{n>=1} (n^2)^n * exp(-n^2*x) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n. - Paul D. Hanna, Oct 17 2012
G.f.: Sum_{n > 0} (a(n)*n!/(2*n)!)*x^n = x*B'(x)/B(x)-1, where B(x) satisfies B(x)^2 = x*(exp(B(x))-1). - Vladimir Kruchinin, Mar 13 2013
a(n) = Sum_{j = 0..n} (-1)^(n-j)*n^j*binomial(2*n,j)*stirling2(2*n-j,n). - Vladimir Kruchinin, Jun 14 2013

Typo in Mathematica program fixed by Vincenzo Librandi, May 04 2013

a(0)=1 prepended by Alois P. Heinz, Feb 01 2018

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

Original entry on oeis.org

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

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

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

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

Seiichi Manyama, Feb 12 2022

nonn

Cf. A007820, A350376, A383862, A384060.

Cf. A351507, A384060.

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

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

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

A383843 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/Product_{j=0..k} (1 - j*x)^2.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 23, 4, 0, 1, 20, 86, 72, 5, 0, 1, 30, 230, 480, 201, 6, 0, 1, 42, 505, 2000, 2307, 522, 7, 0, 1, 56, 973, 6300, 14627, 10044, 1291, 8, 0, 1, 72, 1708, 16464, 65002, 95060, 40792, 3084, 9, 0, 1, 90, 2796, 37632, 227542, 587580, 567240, 157440, 7181, 10, 0

Offset: 0

Seiichi Manyama, May 12 2025

			Square array begins:
  1, 1,    1,     1,      1,       1,        1, ...
  0, 2,    6,    12,     20,      30,       42, ...
  0, 3,   23,    86,    230,     505,      973, ...
  0, 4,   72,   480,   2000,    6300,    16464, ...
  0, 5,  201,  2307,  14627,   65002,   227542, ...
  0, 6,  522, 10044,  95060,  587580,  2725380, ...
  0, 7, 1291, 40792, 567240, 4817990, 29331038, ...

Columns k=0..4 give A000007, A000027(n+1), A045618, A383841, A383842.
Main diagonal gives A350376.
A(n,n-1) gives A383880.
Cf. A106800, A287532.

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

A(n,k) = Sum_{j=0..n} Stirling2(j+k,k) * Stirling2(n-j+k,k).

A383880 a(n) = [x^n] 1/Product_{k=0..n-1} (1 - k*x)^2.

Original entry on oeis.org

1, 0, 3, 72, 2307, 95060, 4817990, 290523576, 20333487251, 1621036680120, 145057745669850, 14399349523416000, 1570425994090538574, 186674663305762642296, 24021930409036829669036, 3327140929951823209016400, 493515678917684006649451651, 78054583374364036172432641200

Offset: 0

Seiichi Manyama, May 13 2025

nonn

Cf. A350376, A383883.

Cf. A342111.

Join[{1}, Table[Sum[StirlingS2[n + k - 1, n - 1]*StirlingS2[2*n - k - 1, n - 1], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 14 2025 *)

a(n) = polcoef(1/prod(k=0, n-1, 1-k*x+x*O(x^n))^2, n);

a(n) = Sum_{k=0..n} Stirling2(n+k-1,n-1) * Stirling2(2*n-k-1,n-1) for n > 0.

a(n) ~ 3^(3*n - 3/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n - 1/2) * exp(n) * (3 - 2*w)^n * w^(2*n - 3/2)), where w = -LambertW(-3*exp(-3/2)/2). - Vaclav Kotesovec, May 14 2025

A383883 a(n) = [x^n] 1/((1 - nx) Product_{k=0..n-1} (1 - k*x)^2).

Original entry on oeis.org

1, 1, 11, 222, 6627, 262570, 12978758, 769079444, 53138842515, 4194648739710, 372421403333850, 36733739199892020, 3985122473105099406, 471598870326072262644, 60456151456891375730860, 8345905345383943433713800, 1234395864446065862689721475, 194738649118647202909304657910

Offset: 0

Seiichi Manyama, May 13 2025

nonn

Cf. A350376, A383880.

Cf. A187235, A287532.

Join[{1}, Table[Sum[StirlingS2[n + k - 1, n - 1]*StirlingS2[2*n - k, n], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 14 2025 *)

a(n) = polcoef(1/((1-n*x)*prod(k=0, n-1, 1-k*x+x*O(x^n))^2), n);

a(n) = Sum_{k=0..n} Stirling2(n+k-1,n-1) * Stirling2(2*n-k,n) for n > 0.
a(n) = A287532(n,n).
a(n) ~ 3^(3*n - 1/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n + 1/2) * exp(n) * (3 - 2*w)^n * w^(2*n - 1/2)), where w = -LambertW(-3*exp(-3/2)/2). - Vaclav Kotesovec, May 14 2025

A384086 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^2.

Original entry on oeis.org

1, 4, 72, 2352, 112000, 7023540, 546991704, 50923706176, 5517464159232, 682067031126660, 94744306830613000, 14610279918692775504, 2476682373835289303424, 457771369968515293229812, 91624876032673265663215800, 19743379886572250897986694400, 4556982707091255612929249419264

Offset: 0

Vaclav Kotesovec, May 19 2025

nonn

Cf. A129256, A350376.

Cf. A350366, A384087, A384088, A351764.

Table[SeriesCoefficient[Product[(1+k*x)^2/(1-k*x)^2, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) ~ c * d^n * n! / n, where d = 15.357995623209995052090556511543938190953157405669200... and c = 0.3746298100044008083790505105262276548713201624206421...

A384206 a(n) = [x^(3n)] Product_{k=0..n} 1/(1 - kx)^2.

Original entry on oeis.org

1, 4, 1291, 2107596, 9822847079, 99559982844000, 1870441451243408425, 58630795546429054116336, 2846132741588198942785663319, 202389763024999232451527049522000, 20194222519959431156536932169706390700, 2731878423936456763814384150978735866605108

Offset: 0

Vaclav Kotesovec, May 22 2025

nonn

Cf. A350376, A384207.

Table[SeriesCoefficient[Product[1/(1-k*x)^2, {k, 0, n}], {x, 0, 3*n}], {n, 0, 15}]
Table[Sum[StirlingS2[i+n, n] * StirlingS2[4*n-i, n], {i, 0, 3*n}], {n, 0, 15}]

a(n) = Sum_{k=0..3*n} Stirling2(n+k, n) * Stirling2(4*n-k, n).

a(n) ~ 5^(5*n + 1/2) * n^(3*n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n + 3/2) * exp(3*n) * w^(2*n + 1/2) * (5 - 2*w)^(3*n)), where w = -LambertW(-5*exp(-5/2)/2) = 0.268388115976977211740078521072609338...

cp's OEIS Frontend

A007820 Stirling numbers of second kind S(2n,n).

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

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A383862 a(n) = [x^n] Product_{k=0..n} 1/(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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A351508 a(n) = [x^n] Product_{k=1..n} 1/(1 - k*x)^n.

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A383843 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/Product_{j=0..k} (1 - j*x)^2.

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A383880 a(n) = [x^n] 1/Product_{k=0..n-1} (1 - k*x)^2.

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A383883 a(n) = [x^n] 1/((1 - n*x) * Product_{k=0..n-1} (1 - k*x)^2).

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A383883 a(n) = [x^n] 1/((1 - nx) Product_{k=0..n-1} (1 - k*x)^2).

A384086 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^2.

A384206 a(n) = [x^(3n)] Product_{k=0..n} 1/(1 - kx)^2.