A187646 -id:A187646 - 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

A285862 Number of permutations of [2n] with n ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Original entry on oeis.org

1, 1, 19, 1005, 62601, 6061545, 868380535, 142349568361, 27564092244689, 6325532235438273, 1673378033771898675, 505141951803309946125, 170002056228253072537065, 63255335047795174479833625, 25805276337820748477042392695, 11427131417576257617280878155625

Offset: 0

Alois P. Heinz, Apr 27 2017

nonn

			a(1) = 1: (12).
a(2) = 19: (123)(4), (4)(123), (132)(4), (4)(132), (124)(3), (3)(124), (142)(3), (3)(142), (134)(2), (2)(134), (143)(2), (2)(143), (1)(234), (234)(1), (1)(243), (243)(1),  (12)(34), (13)(24), (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Permutation

Cf. A187646, A285849, A285926.

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))
    end:
a:= n-> coeff(b(2*n$2, 0), x, n):
seq(a(n), n=0..20);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]]];
a[n_] := Coefficient[b[2n, 2n, 0], x, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2018, from Maple *)

a(n) = A285849(2n,n).

A242676 a(n) = |Stirling1(4*n,n)|.

Original entry on oeis.org

1, 6, 13068, 150917976, 5056995703824, 371384787345228000, 50779532534302850198976, 11616723683566425573507775872, 4123257155075936045020928754053376, 2146734309994687055429549444238169536000, 1569808063009967047226374755685187772671339520

Offset: 0

Vaclav Kotesovec, May 20 2014

Generally, for p>=2 is Abs(StirlingS1(p*n,n)) asymptotic to n^((p-1)*n) * c^(p*n) * p^((2*p-1)*n) / (sqrt(2*Pi*p*(c-1)*n) * exp((p-1)*n) * (c*p-1)^((p-1)*n)), where c = -LambertW(-1,-exp(-1/p)/p).

Cf. A187646, A237993, A217914.

Maple
```
seq(abs(Stirling1(4*n,n)), n=0..20);
```
Mathematica
```
Table[Abs[StirlingS1[4*n, n]],{n,0,20}]
```

a(n) ~ n^(3*n) * c^(4*n) * 2^(14*n-1) / (sqrt(2*Pi*(c-1)*n) * exp(3*n) * (4*c-1)^(3*n)), where c = -LambertW(-1,-exp(-1/4)/4) = 2.58666298226305388118285...
From Seiichi Manyama, May 20 2025: (Start)
a(n) = A132393(4*n,n).
a(n) = (4*n)! * [x^(4*n)] (-log(1 - x))^n / n!. (End)

A187661 Binomial convolution of the (signless) central Stirling numbers of the first kind and the central Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 20, 369, 10192, 379850, 17930697, 1027046517, 69216504576, 5363945384274, 469658243947850, 45827641349686636, 4928867833029014503, 579101340954599901152, 73778702335232336908585, 10129059530832922239925140

Offset: 0

Emanuele Munarini, Mar 12 2011

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A007820, A187646.

seq(sum(binomial(n,k) * abs(combinat[stirling1](2*k, k)) * combinat[stirling2](2*(n-k), n-k), k=0..n), n=0..12);

Table[Sum[Binomial[n, k]Abs[StirlingS1[2k, k]]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 15}]

makelist(sum(binomial(n,k)*abs(stirling1(2*k,k))*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);

a(n) = sum(k=0, n, binomial(n,k)*abs(stirling(2*k, k, 1)*stirling(2*(n-k), n-k, 2))); \\ Michel Marcus, May 28 2017

a(n) = Sum_{k=0..n} binomial(n,k) * s(2*k,k) * S(2*n-2*k,n-k).

a(n) ~ m * n^n * c^(2*n) * 2^(3*n-1) / (sqrt(Pi*(c-1)*n) * exp(n) * (2*c-1)^n), where c = -LambertW(-1,-exp(-1/2)/2) = 1.75643120862616967698..., and m = Sum_{j>=0} StirlingS2(2*j,j) * (2*c-1)^j / (j! * 2^(3*j) * c^(2*j)) = 1.170003674502655133465266152119563086693466... . - Vaclav Kotesovec, May 22 2014

A237993 a(n) = |Stirling1(3*n,n)|.

Original entry on oeis.org

1, 2, 274, 118124, 105258076, 159721605680, 369012649234384, 1206647803780373360, 5304713715525445812976, 30180059720580991603896800, 215760462268683520394805979744, 1893448925578239663637174767335168, 20012008248418194052035539503977759232

Offset: 0

Vaclav Kotesovec, May 20 2014

Cf. A132393, A187646, A242676, A217913.

Maple
```
seq(abs(Stirling1(3*n,n)), n=0..20);
```
Mathematica
```
Table[Abs[StirlingS1[3*n, n]],{n,0,20}]
```

a(n) ~ n^(2*n) * c^(3*n) * 3^(5*n) / (sqrt(6*Pi*(c-1)*n) * exp(2*n) * (3*c-1)^(2*n)), where c = -LambertW(-1,-exp(-1/3)/3) = 2.237147027773716818...
From Seiichi Manyama, May 20 2025: (Start)
a(n) = A132393(3*n,n).
a(n) = (3*n)! * [x^(3*n)] (-log(1 - x))^n / n!. (End)

A348023 a(n) = [x^n] Product_{k=0..2n-1} (x + (-1)^k k).

Original entry on oeis.org

1, -1, -5, 51, 1009, -17765, -636385, 15875083, 828730833, -26734633497, -1837389309405, 72689703927555, 6202163408816881, -290609121018808813, -29631450442239587305, 1604436906627413556075, 190333687041180914380065, -11692931012450127257939505, -1582217617564533519968758645

Offset: 0

Seiichi Manyama, Sep 25 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..351

Cf. A047991, A187646, A348024.

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

a(n) = A047991(2*n-1,n) for n>0.

A383882 a(n) = [x^n] Product_{k=1..4n} 1/(1 - kx).

Original entry on oeis.org

1, 10, 750, 106470, 22350954, 6220194750, 2157580085700, 896587036640680, 434225240080346858, 240175986308550372366, 149377949042637543000150, 103192471874508023383125750, 78394850841083734162487127720, 64957213308036504429927388238088, 58298851680969051596827194829579744

Offset: 0

Vaclav Kotesovec, May 13 2025

nonn

In general, for m>=1, Stirling2((m+1)*n, m*n) ~ (-1)^(m*n) * (m+1)^((m+1)*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w(m))) * exp(n) * m^(m*n + 1/2) * w(m)^(m*n) * (1 + 1/m + w(m))^n), where w(m) = LambertW(-(1 + 1/m)/exp(1 + 1/m)).

Cf. A007820 (m=1), A348084 (m=2), A383881 (m=3).
Cf. A217913.
Cf. A187646, A384129, A384130.

Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 4*n}], {x, 0, n}], {n, 0, 15}]
Table[StirlingS2[5*n, 4*n], {n, 0, 15}]
Table[SeriesCoefficient[1/(Pochhammer[1 - 1/x, 4*n]*x^(4*n)), {x, 0, n}], {n, 0, 15}]

a(n) = Stirling2(5*n,4*n).

a(n) ~ 5^(5*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 4^(4*n + 1/2) * w^(4*n) * (5/4 + w)^n), where w = LambertW(-5/(4*exp(5/4))).

A384129 Number of permutations of 3n objects with exactly 2n cycles.

Original entry on oeis.org

1, 3, 85, 4536, 357423, 37312275, 4853222764, 756111184500, 137272511800831, 28460103232088385, 6634460278534540725, 1717750737160208150400, 489078062391738506912340, 151874660255802127280374140, 51082995429153110239690350120, 18500755859447038660174079965500

Offset: 0

Seiichi Manyama, May 20 2025

Cf. A132393, A187646, A348084.

PARI
```
a(n) = abs(stirling(3*n, 2*n, 1));
```

a(n) = A132393(3*n,2*n) = |Stirling1(3*n,2*n)|.
a(n) = (3*n)! * [x^(3*n)] log(1 - x)^(2*n) / (2*n)!.
a(n) ~ 3^(4*n - 1/2) * w^(3*n) * n^(n - 1/2) / (sqrt(Pi*(w-1)) * 2^(2*n + 1/2) * exp(n) * (3*w-2)^n), where w = -LambertW(-1, -2*exp(-2/3)/3) = 1.4293552275170056487... - Vaclav Kotesovec, May 23 2025

A384130 Number of permutations of 4n objects with exactly 3n cycles.

Original entry on oeis.org

1, 6, 322, 32670, 4899622, 973941900, 241276443496, 71603372991150, 24764667228756390, 9781650150525639540, 4344363139637533397580, 2143082171052546774398348, 1162585907585797437278546956, 687872810620417599693839111880, 440840269604491448260396623711300

Offset: 0

Seiichi Manyama, May 20 2025

In general, for m>=1, abs(Stirling1((m+1)*n, m*n)) ~ (m+1)^((m+2)*n - 1/2) * w(m)^((m+1)*n) * n^(n - 1/2) / (sqrt(2*Pi*(w(m)-1)) * exp(n) * m^(m*n) * ((m+1)*w(m) - m)^n), where w(m) = -LambertW(-1, -m*exp(-m/(m+1))/(m+1)). - Vaclav Kotesovec, May 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A132393, A242676, A383881, A383882.

Cf. A187646, A384129.

[&+[Abs(StirlingFirst(4*n, 3*n))]: n in [0..15]]; // Vincenzo Librandi, May 21 2025

a[n_]:=Abs[StirlingS1[4 n,3 n]] Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 21 2025 *)

PARI
```
a(n) = abs(stirling(4*n, 3*n, 1));
```

a(n) = A132393(4*n,3*n) = |Stirling1(4*n,3*n)|.
a(n) = (4*n)! * [x^(4*n)] (-log(1 - x))^(3*n) / (3*n)!.
a(n) ~ 2^(10*n - 3/2) * n^(n - 1/2) * w^(4*n) / (sqrt(Pi*(w-1)) * 3^(3*n) * exp(n) * (4*w-3)^n), where w = -LambertW(-1, -3*exp(-3/4)/4) = 1.3002007416590685881... - Vaclav Kotesovec, May 23 2025

A154415 The middle Stirling numbers of first kind: a(n) = Stirling1(n, floor(n/2)).

Original entry on oeis.org

1, 0, -1, 2, 11, -50, -225, 1624, 6769, -67284, -269325, 3416930, 13339535, -206070150, -790943153, 14409322928, 54631129553, -1146901283528, -4308105301929, 102417740732658, 381922055502195, -10142299865511450

Offset: 0

Roger L. Bagula, Jan 09 2009

sign

The signless central Stirling numbers of the first kind Stirling1(2*n, n) are A187646.

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A048994, A187646.

Table[StirlingS1[n, Floor[n/2]], {n, 0, 30}]

a(n) = stirling(n, n\2, 1); \\ Michel Marcus, Sep 16 2016

Name clarified by Peter Luschny, Jan 06 2020

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

PARI

PARI

Python

Sage

Formula

Extensions

A285862 Number of permutations of [2n] with n ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A242676 a(n) = |Stirling1(4*n,n)|.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

A187661 Binomial convolution of the (signless) central Stirling numbers of the first kind and the central Stirling numbers of the second kind.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A237993 a(n) = |Stirling1(3*n,n)|.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A348023 a(n) = [x^n] Product_{k=0..2*n-1} (x + (-1)^k * k).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A383882 a(n) = [x^n] Product_{k=1..4*n} 1/(1 - k*x).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A348023 a(n) = [x^n] Product_{k=0..2n-1} (x + (-1)^k k).

A383882 a(n) = [x^n] Product_{k=1..4n} 1/(1 - kx).

A384129 Number of permutations of 3n objects with exactly 2n cycles.

A384130 Number of permutations of 4n objects with exactly 3n cycles.