A007820 -id:A007820 - cp's OEIS frontend

A293609 Triangle read by rows, a refinement of the central Stirling numbers of the first kind A187646, T(n, k) for n >= 0 and 0 <= k <= n.

1, 1, 0, 7, 4, 0, 90, 120, 15, 0, 1701, 3696, 1316, 56, 0, 42525, 129780, 84630, 12180, 210, 0, 1323652, 5233404, 5184894, 1492744, 104049, 792, 0, 49329280, 240240000, 326426100, 151251100, 22840818, 852852, 3003, 0

Offset: 0

Peter Luschny, Oct 15 2017

			Triangle starts:
[0]        1
[1]        1,         0
[2]        7,         4,         0
[3]       90,       120,        15,         0
[4]     1701,      3696,      1316,        56,        0
[5]    42525,    129780,     84630,     12180,      210,      0
[6]  1323652,   5233404,   5184894,   1492744,   104049,    792,    0
[7] 49329280, 240240000, 326426100, 151251100, 22840818, 852852, 3003, 0

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Row sums are A187646. T(n, 0) = A007820(n) the central Stirling numbers of the second kind A048993. T(n, n-1) = A001791(n) for n>=1.

Cf. A293616.

for n in [$0..9] do seq(A293616(n, n, k), k=0..n) od;

A293609Row[n_] := If[n==0, {1}, Join[CoefficientList[x^(-n) (1 - x)^(2n) PolyLog[-2n, n, x] /. Log[1 - x] -> 0, x], {0}]];
Table[A293609Row[n], {n, 0, 7}] // Flatten

T(n, k) = A293616(n, n, k) for k = 0..n. The main diagonal in terms of rows (!) of the array of triangles A293616. T_row(n) is row n of triangle A293616(n,.,.), i.e. T_row(0) = [1] is row 0 of A000007, T_row(1) = [1, 0] is row 1 of A173018, T_row(2) = [7, 4, 0] is row 2 of A062253, and so on.

Let h(n) = x^(-n)*(1 - x)^(2*n)*PolyLog(-2*n, n, x) and p(n) the polynomial given by the expansion of h(n) after replacing log(1 - x) by 0. Then T(n, k) is the k-th coefficient of p(n) for 0 <= k < n.

A293926 Triangle read by rows, T(n, k) = Pochhammer(n, k) * Stirling2(2*n, k + n) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 7, 12, 6, 90, 195, 180, 60, 1701, 4200, 5320, 3360, 840, 42525, 114135, 176400, 157500, 75600, 15120, 1323652, 3764376, 6679134, 7484400, 5155920, 1995840, 332640, 49329280, 146386240, 287567280, 379387008, 332972640, 186666480, 60540480, 8648640

Offset: 0

Peter Luschny, Oct 22 2017

			Triangle starts:
[0]       1
[1]       1,       1
[2]       7,      12,       6
[3]      90,     195,     180,      60
[4]    1701,    4200,    5320,    3360,     840
[5]   42525,  114135,  176400,  157500,   75600,   15120
[6] 1323652, 3764376, 6679134, 7484400, 5155920, 1995840, 332640

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T(n,0) = Stirling2(2*n,n) = A007820(n), T(n,n) = A000407(n).

Cf. A293617.

A293926 := (n, k) -> A293617(n, n, k ):
seq(seq(A293926(n, k), k=0..n), n=0..7);

A293617[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
A293926Row[n_] := Table[A293617[n, n, k], {k, 0, n}];
Table[A293926Row[n], {n, 0, 7}] // Flatten

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, ((n+k-1)!/(n-1)!)*stirling(2*n, n + k, 2)), ", "))) \\ G. C. Greubel, Nov 19 2017

T(n, k) = A293617(n, n, k).

A324241 Number of set partitions of [2n] where each subset is again partitioned into n nonempty subsets.

Original entry on oeis.org

1, 2, 10, 100, 1736, 42651, 1324114, 49330996, 2141770488, 106175420065, 5917585057033, 366282501223002, 24930204592110338, 1850568574258750360, 148782988064395367700, 12879868072770703598760, 1194461517469808134322280, 118144018577011379763287565

Offset: 0

Alois P. Heinz, Sep 02 2019

nonn

			a(2) = 10: 123/4, 124/3, 12/34, 134/2, 13/24, 14/23, 1/234, 1/2|3/4, 1/3|2/4, 1/4|2/3.

Alois P. Heinz, Table of n, a(n) for n = 0..345
Wikipedia, Partition of a set

Cf. A007820, A088218, A324162.

b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)
      *binomial(n-1, j-1)*Stirling2(j, k), j=k..n))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..18);

b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 0 || k > n, 0, Sum[b[n-j, k]* Binomial[n - 1, j - 1] StirlingS2[j, k], {j, k, n}]]];
a[n_] := b[2n, n];
a /@ Range[0, 18] (* Jean-François Alcover, May 05 2020, after Maple *)

a(n) = if(n==0, 1, stirling(2*n, n, 2)+binomial(2*n, n)/2); \\ Seiichi Manyama, May 08 2022

a(n) = A324162(2n,n).

a(n) = A007820(n) + A088218(n) for n > 0. - Seiichi Manyama, May 08 2022

A354797 Triangle read by rows. T(n, k) = |Stirling1(n, k)| * Stirling2(n + k, n) = A132393(n, k) * A048993(n + k, n).

Original entry on oeis.org

1, 0, 1, 0, 3, 7, 0, 12, 75, 90, 0, 60, 715, 2100, 1701, 0, 360, 7000, 36750, 69510, 42525, 0, 2520, 72884, 595350, 1940295, 2692305, 1323652, 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280, 0, 181440, 9801000, 156008160, 1076453763, 3723239520, 6733767040, 6065579520, 2141764053

Offset: 0

Peter Luschny, Jun 06 2022

			Table T(n, k) begins:
[0] 1
[1] 0,     1
[2] 0,     3,      7
[3] 0,    12,     75,      90
[4] 0,    60,    715,    2100,     1701
[5] 0,   360,   7000,   36750,    69510,     42525
[6] 0,  2520,  72884,  595350,  1940295,   2692305,   1323652
[7] 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280

Mike Earnest, Counting endofunctions by inclusion-exclusion, at Math.StackExchange.

Cf. A048993, A001710, A007820, A132393, A000312, A354977.

T := (n, k) -> abs(Stirling1(n, k))*Stirling2(n + k, n):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;

Sum_{k=0..n} (-1)^(n - k)*T(n, k) = n^n. - Werner Schulte, Jun 03 2022 in A000312. [Formerly a conjecture, now proved by Mike Earnest, see link.]

T(n, k) = A132393(n, k) * A354977(n, k) = (1/n!) * Sum_{j=0..n} (-1)^(j + k) * binomial(n, j) * Stirling1(n, k) * j^(n + k).

A354977 Triangle read by rows. T(n, k) = Sum_{j=0..n}((-1)^(n-j)binomial(n, j)j^(n+k)) / n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 1, 6, 25, 90, 1, 10, 65, 350, 1701, 1, 15, 140, 1050, 6951, 42525, 1, 21, 266, 2646, 22827, 179487, 1323652, 1, 28, 462, 5880, 63987, 627396, 5715424, 49329280, 1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053

Offset: 0

Peter Luschny, Jun 15 2022

			Triangle T(n, k) begins:
[0] 1;
[1] 1,  1;
[2] 1,  3,   7;
[3] 1,  6,  25,    90;
[4] 1, 10,  65,   350,   1701;
[5] 1, 15, 140,  1050,   6951,   42525;
[6] 1, 21, 266,  2646,  22827,  179487,  1323652;
[7] 1, 28, 462,  5880,  63987,  627396,  5715424,  49329280;
[8] 1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053;

Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 13.

T(n,1) = A000217, T(n,n) = A007820, A354978 (row sums), A048993.

T := (n, k) -> add((-1)^(n - j)*binomial(n, j)*j^(n + k), j = 0..n) / n!:
seq(seq(T(n, k), k = 0..n), n = 0..8);

T(n, k) = Stirling2(n + k, n).

A383869 a(n) = [x^n] 1/Product_{k=0..n} (1 - (n+k)*x).

Original entry on oeis.org

1, 3, 55, 1890, 95781, 6427575, 537306484, 53791898160, 6275077781973, 835898091070185, 125195263380478655, 20825548503275385870, 3809430011164368694260, 759987002381075483922180, 164221938436980055710082200, 38209754165858724861944820000, 9524153723280871205135022364485

Offset: 0

Seiichi Manyama, May 13 2025

nonn

Central terms of triangle A143395.

Cf. A007820, A129506.

Table[SeriesCoefficient[1/Product[(1 - (n + k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 17 2025 *)

a(n) = sum(k=0, n, (-1)^(n-k)*(n+k)^(2*n)*binomial(n, k))/n!;

a(n) = (1/n!) * Sum_{k=0..n} (-1)^(n-k) * (n+k)^(2*n) * binomial(n,k).
a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*n,n+k) * Stirling2(n+k,n).
a(n) = Sum_{k=0..n} (-1)^k * (2*n)^(n-k) * binomial(2*n,n+k) * Stirling2(n+k,n).
a(n) ~ (r-1)^((r-1)*n) * (1+r)^(2*n + 1) * exp(n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + (4-r)*r)) * r^(r*n)), where r = 2.106565648173949260853515992430777519716829316322... is the root of the equation exp(2/(1+r)) = r/(r-1). - Vaclav Kotesovec, May 17 2025

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))).

A187647 Partial sums of the central Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 9, 99, 1800, 44325, 1367977, 50697257, 2192461310, 108367857065, 6025952821720, 372308453692006, 25302513044450266, 1875871087298000326, 150658859151673309726, 13030526931922299349726, 1207492044401730133131811

Offset: 0

Emanuele Munarini, Mar 12 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..49

Cf. A007820, A226775.

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

Table[Sum[StirlingS2[2k, k], {k, 0, n}], {n, 0, 16}]

makelist(sum(stirling2(2*k,k),k,0,n),n,0,12);

a(n) = sum_{k=0..n} A048993(2*k,k).
a(n+1)-a(n) = A007820(n+1).
a(n) ~ n^n * 2^(2*n) / (sqrt(2*Pi*(1-c)*n) * exp(n) * (2-c)^n * c^n), where c = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, May 11 2014

A218671 O.g.f.: Sum_{n>=0} n^(2n) (1+nx)^n x^n/n! * exp(-n^2x(1+n*x)).

Original entry on oeis.org

1, 1, 8, 120, 2635, 76503, 2764957, 119634152, 6030195490, 347037131298, 22453144758980, 1613322276606404, 127466755375275614, 10983423290600347408, 1025046637630590359928, 103004615955568528609200, 11088429267977228122393005, 1273093489376335864500416685

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare g.f. to the curious identity:

1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).

			O.g.f.: A(x) = 1 + x + 8*x^2 + 120*x^3 + 2635*x^4 + 76503*x^5 +...
where
A(x) = 1 + (1+x)*x*exp(-x*(1+x)) + 2^4*(1+2*x)^2*x^2/2!*exp(-2^2*x*(1+2*x)) + 3^6*(1+3*x)^3*x^3/3!*exp(-3^2*x*(1+3*x)) + 4^8*(1+4*x)^4*x^4/4!*exp(-4^2*x*(1+4*x)) + 5^10*(1+5*x)^5*x^5/5!*exp(-5^2*x*(1+5*x)) +...
simplifies to a power series in x with integer coefficients.

Cf. A007820, A218670, A218672.

{a(n)= my(A=sum(k=0, n, k^(2*k)*(1+k*x)^k*x^k/k!*exp(-k^2*x*(1+k*x)+x*O(x^n)))); polcoef(A, n)}
for(n=0,30,print1(a(n),", "))

A220282 E.g.f.: 1/(1-x) = Sum_{n>=0} a(n) * exp(-n^2x) x^n/n!.

Original entry on oeis.org

1, 1, 4, 51, 1480, 79765, 7010496, 920281831, 169526669824, 41844075277545, 13357347571244800, 5362349333225289691, 2646862288162043664384, 1576780272924188221429501, 1116120717235502072828661760, 926421799655193830945493519375, 891516461371835173578650979598336

Offset: 0

Paul D. Hanna, Dec 11 2012

nonn

Compare to the identity: 1/(1-x) = Sum_{n>=0} n^n * exp(-n*x) * x^n/n!.
Compare to the o.g.f. of A007820:
Sum_{n>=0} S2(2*n,n)*x^n = Sum_{n>=0} (n^2)^n * exp(-n^2*x) * x^n/n!.

			E.g.f.: 1/(1-x) = 1 + 1*exp(-x)*x + 4*exp(-2^2*x)*x^2/2! + 51*exp(-3^2*x)*x^3/3! + 1480*exp(-4^2*x)*x^4/4! + 79765*exp(-5^2*x)*x^5/5! + 7010496*exp(-6^2*x)*x^6/6!+...

Cf. A007820, A210029.

{a(n)=n!*polcoeff(1/(1-x+x*O(x^n))-sum(k=0,n-1,a(k)*x^k/k!*exp(-k^2*x+x*O(x^n))), n)}
for(n=0,16,print1(a(n),", "))

cp's OEIS Frontend

A293609 Triangle read by rows, a refinement of the central Stirling numbers of the first kind A187646, T(n, k) for n >= 0 and 0 <= k <= n.

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A293926 Triangle read by rows, T(n, k) = Pochhammer(n, k) * Stirling2(2*n, k + n) for n >= 0 and 0 <= k <= n.

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A324241 Number of set partitions of [2n] where each subset is again partitioned into n nonempty subsets.

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A354797 Triangle read by rows. T(n, k) = |Stirling1(n, k)| * Stirling2(n + k, n) = A132393(n, k) * A048993(n + k, n).

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A354977 Triangle read by rows. T(n, k) = Sum_{j=0..n}((-1)^(n-j)*binomial(n, j)*j^(n+k)) / n!.

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A383869 a(n) = [x^n] 1/Product_{k=0..n} (1 - (n+k)*x).

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

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A187647 Partial sums of the central Stirling numbers of the second kind.

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A354977 Triangle read by rows. T(n, k) = Sum_{j=0..n}((-1)^(n-j)binomial(n, j)j^(n+k)) / n!.

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

A218671 O.g.f.: Sum_{n>=0} n^(2n) (1+nx)^n x^n/n! * exp(-n^2x(1+n*x)).

A220282 E.g.f.: 1/(1-x) = Sum_{n>=0} a(n) * exp(-n^2x) x^n/n!.