A001861 -id:A001861 - cp's OEIS frontend

A335980 Expansion of e.g.f. exp(2 * (1 - exp(-x)) + x).

1, 3, 7, 11, 7, -5, 23, 75, -281, -101, 4663, -14229, -41721, 532667, -1464489, -8840053, 103689511, -313202725, -2348557705, 32041266859, -127039882425, -762423051013, 14393151011735, -81523161874741, -236027974047897, 8564406463119387

Offset: 0

Ilya Gutkovskiy, Jul 03 2020

sign

Cf. A001861, A035009, A109747, A213170, A335868, A335981, A335982.

nmax = 25; CoefficientList[Series[Exp[2 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 2 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

my(N=33, x='x+O('x^N)); Vec(serlaplace(exp(2 * (1 - exp(-x)) + x))) \\ Joerg Arndt, Jul 04 2020

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

a(0) = 1; a(n) = a(n-1) + 2 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).

A352617 Expansion of e.g.f. exp( exp(x) + sinh(x) - 1 ).

Original entry on oeis.org

1, 2, 5, 16, 60, 254, 1199, 6206, 34827, 210264, 1355992, 9288954, 67279309, 513149498, 4107383185, 34398823888, 300629113292, 2735356900806, 25857446103571, 253472859754918, 2572266378189583, 26981781750668760, 292136508070103208, 3260640536587635410, 37472102225288489529

Offset: 0

Ilya Gutkovskiy, Mar 24 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..565

Cf. A000110, A001861, A003724, A005046, A352279, A352326.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n-1, k-1)*(1+(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Mar 24 2022

nmax = 24; CoefficientList[Series[Exp[Exp[x] + Sinh[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 1, k - 1] (3 - (-1)^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

my(x='x+O('x^30)); Vec(serlaplace(exp( exp(x) + sinh(x) - 1 ))) \\ Michel Marcus, Mar 24 2022

a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n-1,k-1) * (3 - (-1)^k) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * A000110(k) * A003724(n-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A005046(k) * A352279(n-2*k).

A355252 Expansion of e.g.f. exp(2(exp(x) - 1) + 3x).

Original entry on oeis.org

1, 5, 27, 157, 979, 6517, 46107, 345261, 2726243, 22623525, 196712171, 1787356765, 16929897395, 166808851541, 1706299041211, 18088031239437, 198392625389315, 2248104026019461, 26283054263021963, 316637825898555069, 3926250785070282579, 50056384077880370101

Offset: 0

Vaclav Kotesovec, Jun 26 2022

nonn

Binomial transform of A355247.

Vaclav Kotesovec, Table of n, a(n) for n = 0..550

Cf. A001861, A035009, A355247, A355253.

nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+3*x], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1) + 3*x))) \\ Michel Marcus, Dec 04 2023

a(n) ~ n^(n+3) * exp(n/LambertW(n/2) - n - 2) / (8 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+3)).

a(0) = 1; a(n) = 3 * a(n-1) + 2 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023

A003466 Number of minimal covers of an n-set that have exactly one point which appears in more than one set in the cover.

Original entry on oeis.org

0, 3, 28, 210, 1506, 10871, 80592, 618939, 4942070, 41076508, 355372524, 3198027157, 29905143464, 290243182755, 2920041395248, 30414515081650, 327567816748638, 3643600859114439, 41809197852127240, 494367554679088923, 6017481714095327410

Offset: 2

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 2..510
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.

Cf. A046165.

Column k=1 of A282575.

seq(n*add((2^k-k-1)*Stirling2(n-1,k),k=1..n-1), n = 2 .. 30); # Robert Israel, May 21 2015

nn = 20; Range[0, nn]! CoefficientList[Series[Sum[ (Exp[x] - 1)^n/n! (2^n - n - 1) x, {n, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Feb 18 2017 *)
a[2]=0;a[3]=3;a[4]=28;a[n_]:=n*Sum[(2^k-k-1)* StirlingS2[n-1,k], {k,1,n-1}];Table[a[n],{n,2,22}] (* Indranil Ghosh, Feb 20 2017 *)

a(n) = n * Sum_{k=1..n-1} (2^k-k-1) * S2(n-1,k) where S2(n,k) are the Stirling numbers of the second kind. - Sean A. Irvine, May 20 2015

a(n) = n * (A001861(n-1) - A005493(n-2) - A000110(n-1)). - Robert Israel, May 21 2015

More terms from Sean A. Irvine, May 20 2015

Title clarified by Geoffrey Critzer, Feb 18 2017

A036076 Expansion of e.g.f. exp((exp(p*x)-p-1)/p+exp(x)) for p=6.

Original entry on oeis.org

1, 2, 11, 87, 844, 9599, 125545, 1854234, 30407763, 546409567, 10654642428, 223763443039, 5030118977041, 120393730088818, 3054106291046267, 81792080931311015, 2304639285452820684, 68117438479292896255

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Robert Israel, Table of n, a(n) for n = 0..437
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074, ...

egf:=  exp((exp(6*x)-6-1)/6+exp(x)):
S:= series(egf,x,501):
seq(coeff(S,x,i)*i!, i=0..20); # Robert Israel, Nov 27 2022

mx = 16; p = 6; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 6^k * BellB[k, 1/6] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=6. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (6*n/LambertW(6*n))^n * exp(n/LambertW(6*n) + (6*n/LambertW(6*n))^(1/6) - n - 7/6) / sqrt(1 + LambertW(6*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

A036081 The number of partitions of {1..(11n)} that are invariant under a permutation consisting of n 11-cycles.

Original entry on oeis.org

1, 2, 16, 202, 3044, 52794, 1055260, 24081754, 615896308, 17347970202, 531721375308, 17595339114554, 624882463734756, 23691503493287738, 954301756159098172, 40665568780962213530, 1826521141853468785364

Offset: 0

N. J. A. Sloane

nonn

Original name: Sorting numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074.

Column 11 of A162663.

u[0, j_] := 1; u[k_, j_] := u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 11], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 - after Wouter Meeussen in similar sequences *)
mx = 16; p = 11; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 11^k * BellB[k, 1/11] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=11.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=11. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (11*n/LambertW(11*n))^n * exp(n/LambertW(11*n) + (11*n/LambertW(11*n))^(1/11) - n - 12/11) / sqrt(1 + LambertW(11*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 10, 12, 6, 15, 37, 62, 60, 24, 52, 151, 320, 450, 360, 120, 203, 674, 1712, 3120, 3720, 2520, 720, 877, 3263, 9604, 21336, 33600, 34440, 20160, 5040, 4140, 17007, 56674, 147756, 287784, 394800, 352800, 181440, 40320, 21147, 94828

Offset: 0

Vladeta Jovovic, Jan 02 2001

The transpose of this lower unitriangular array is the U factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20L%20factor%20is%20A049020%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The L factor is A049020 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
  [0] [ 1]
  [1] [ 1,    1]
  [2] [ 2,    3,    2]
  [3] [ 5,   10,   12,    6]
  [4] [15,   37,   62,   60,   24]
  [5] [52,  151,  320,  450,  360,  120]
  [6] [203, 674, 1712, 3120, 3720, 2520, 720]
  ...;
E.g.f. for T(n, 2) = (exp(x)-1)^2*(exp(exp(x)-1)) = x^2 + 2*x^3 + 31/12*x^4 + 8/3*x^5 + 107/45*x^6 + 343/180*x^7 + 28337/20160*x^8 + 349/360*x^9 + ...;
E.g.f. for T(n, 3) = (exp(x)-1)^3*(exp(exp(x)-1)) = x^3 + 5/2*x^4 + 15/4*x^5 + 13/3*x^6 + 127/30*x^7 + 1759/480*x^8 + 34961/12096*x^9 + ...

Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

Cf. A000110(n) = T(n,0), A005493(n) = T(n,1), A059099 (row sums).

Cf. A049020, A001861, A046716.

T := proc(n, k) option remember; `if`(k < 0 or k > n, 0,
      `if`(n = 0, 1, k*T(n-1, k-1) + (k+1)*T(n-1, k) + T(n-1, k+1)))
    end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..15); # Peter Bala, Oct 15 2023

E.g.f. for T(n, k): (exp(x)-1)^k*(exp(exp(x)-1)).
n-th row is M^n*[1,0,0,0,...], where M is a tridiagonal matrix with all 1's in the superdiagonal, (1,2,3,...) in the main and subdiagonals; and the rest zeros. - Gary W. Adamson, Jun 23 2011
T(n, k) = k!*A049020(n, k). - R. J. Mathar, May 17 2016
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j). - Peter Luschny, Dec 06 2023

A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 8, 100, 1664, 34336, 843776, 24046912, 779780096, 28357004800, 1143189536768, 50612287301632, 2441525866790912, 127479926768287744, 7163315850315825152, 431046122080208896000, 27655699473265974050816, 1884658377677216933085184

Offset: 1

Johannes W. Meijer, Jul 27 2016

nonn

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.
To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the inverse multinomial transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.
The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.
Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = (1/0!) * (1*x(1))
a(2) = (1/1!) * (1*x(2) - x(1)^2)
a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
a(4) = (1/3!) * (1*x(4) - 4*x(3)*x(1) - 3*x(2)^2 + 12*x(2)*x(1)^2 - 6*x(1)^4)
a(5) = (1/4!) * (1* x(5) - 5*x(4)*x(1) - 10*x(3)*x(2) + 20*x(3)*x(1)^2 + 30*x(2)^2*x(1) -60*x(2)*x(1)^3 + 24*x(1)^5)

Richard P. Feynman, QED, The strange theory of light and matter, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.
Wikipedia, Feynman diagram

Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.

nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: c:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*c(k), k=1..n-1)/n end: a := proc(n): c(n)/(n-1)! end: seq(a(n), n=1..nmax); # End first IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := log(1+add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(exp(add(t(n)*x^n/n, n=1..nmax)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): t(1):= b(1): for n from 2 to nmax+1 do t(n) := solve(d(n)-b(n), t(n)): a(n):=t(n): od: seq(a(n), n=1..nmax); # End third IML program.

nMax = 22; b[n_] := ((2*n-1)!!)^2; c[n_] := c[n] = b[n] - Sum[k*Binomial[n, k]*b[n-k]*c[k], {k, 1, n-1}]/n; a[n_] := c[n]/(n-1)!; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

a(n) = c(n)/(n-1)! with c(n) = b(n) - Sum_{k=1..n-1}(k*binomial(n, k)*b(n-k)*c(k)), n >= 1 and a(0) = undefined, with b(n) = A001818(n) = ((2*n-1)!!)^2.

a(n) = A000079(n-1) * A005411(n), n >= 1.

A308543 Expansion of e.g.f. exp(2(exp(2x) - 1)).

Original entry on oeis.org

1, 4, 24, 176, 1504, 14528, 155520, 1819392, 23019008, 312413184, 4518705152, 69279690752, 1120856170496, 19062628335616, 339681346551808, 6323658075340800, 122680376836358144, 2474677219852288000, 51799971194270646272, 1123121391647711035392

Offset: 0

Ilya Gutkovskiy, Jun 06 2019

nonn

Cf. A000079, A000110, A001861, A002866, A004211, A055882.

nmax = 19; CoefficientList[Series[Exp[2 (Exp[2 x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Sum[4^k x^k/Product[(1 - 2 j x), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[2^(k + 1) Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]
Table[2^n BellB[n, 2], {n, 0, 19}]

O.g.f.: Sum_{k>=0} 4^k*x^k / Product_{j=1..k} (1 - 2*j*x).
E.g.f.: exp(4*exp(x)*sinh(x)).
E.g.f.: g(g(x) - 1), where g(x) = e.g.f. of A000079 (powers of 2).
E.g.f.: f(x)^4, where f(x) = e.g.f. of A004211 (shifts one place left under 2nd-order binomial transform).
a(0) = 1; a(n) = Sum_{k=1..n} 2^(k+1)*binomial(n-1,k-1)*a(n-k).
a(n) = Sum_{k=0..n} 2^(n+k)*Stirling2(n,k).
a(n) = exp(-2) * Sum_{k>=0} 2^(n+k)*k^n/k!.
a(n) = 2^n * A001861(n).

A352624 Expansion of e.g.f. exp(exp(x) + cosh(x) - 2).

Original entry on oeis.org

1, 1, 3, 8, 31, 122, 579, 2886, 16139, 95358, 611111, 4128830, 29709695, 224400022, 1785322699, 14841968646, 129015458195, 1167021383902, 10979895178511, 107113768171950, 1082508179141031, 11308614423992102, 121995294474174963, 1356835055606851286, 15542964081299602811

Offset: 0

Ilya Gutkovskiy, Mar 24 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..566

Cf. A000110, A000807, A001861, A003724, A005046, A352327, A352617.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n-1, k-1)*(2-(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Mar 24 2022

nmax = 24; CoefficientList[Series[Exp[Exp[x] + Cosh[x] - 2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 1, k - 1] (3 + (-1)^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n-1,k-1) * (3 + (-1)^k) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A005046(k) * A000110(n-2*k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A000807(k) * A003724(n-2*k).

cp's OEIS Frontend

A335980 Expansion of e.g.f. exp(2 * (1 - exp(-x)) + x).

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A352617 Expansion of e.g.f. exp( exp(x) + sinh(x) - 1 ).

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A355252 Expansion of e.g.f. exp(2*(exp(x) - 1) + 3*x).

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A003466 Number of minimal covers of an n-set that have exactly one point which appears in more than one set in the cover.

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A036076 Expansion of e.g.f. exp((exp(p*x)-p-1)/p+exp(x)) for p=6.

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Maple

Mathematica

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A036081 The number of partitions of {1..(11n)} that are invariant under a permutation consisting of n 11-cycles.

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Mathematica

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A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

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Maple

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A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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A355252 Expansion of e.g.f. exp(2(exp(x) - 1) + 3x).

A308543 Expansion of e.g.f. exp(2(exp(2x) - 1)).