A091752 -id:A091752 - cp's OEIS frontend

A091753 Fourth column (m=5) of array A091752 ((-1,2)Stirling2) divided by -6.

1, 60, 6720, 1232000, 336336000, 128076748800, 64892219392000, 42217023873024000, 34301331896832000000, 34042166278055936000000, 40523794737397786214400000, 56991191326140341157888000000, 93484550838645539612655616000000, 176901534663898482651640627200000000

Offset: 3

Wolfdieter Lang, Feb 27 2004

Cf. A052502, A091752.

a[n_] := ((n-1)*(n-2)/4) * (3*(n-2))!/((3^(n-2))*(n-2)!); Array[a, 20, 3] (* Amiram Eldar, Sep 01 2025 *)

a(n) = -A091752(n, 5)/6, n>=3.
a(n) = ((n-1)*(n-2)/4)*(3*(n-2))!/((3^(n-2))*(n-2)!) = ((n-1)*(n-2)/4)*A052502(n-2), n>=3.
G.f.: (x^3)*hypergeom([4/3, 5/3, 3], [], 9*x).
a(n) ~ 3^(2*n-7/2) * n^(2*n-2) / (4 * exp(2*n-5/n)). - Amiram Eldar, Sep 01 2025

A091754 Fifth column (m=6) of array A091752 ((-1,2)Stirling2).

Original entry on oeis.org

1, 80, 9520, 1786400, 493292800, 189065676800, 96179539456000, 62739188255744000, 51070871935283200000, 50753775178192486400000, 60478693661116393062400000, 85121458839683971088384000000, 139713174879733993267265536000000, 264509913735543445488643604480000000

Offset: 3

Wolfdieter Lang, Feb 27 2004

Cf. A052502, A091752.

a[n_] := ((3*(n-2))!/((3^(n-2))*(n-2)!))*(9*n^2-27*n+12)/4!; Array[a, 20, 3] (* Amiram Eldar, Sep 01 2025 *)

a(n) = A091752(n, 6), n>=3.
a(n) = ((3*(n-2))!/((3^(n-2))*(n-2)!))*(9*n^2-27*n+12)/4! = A052502(n-2)*(9*n^2-27*n+12)/4!, n>=3.
a(n) ~ (3/e)^(2*n-5/2) * n^(2*n-2) / 8. - Amiram Eldar, Sep 01 2025

A091755 Sixth column (m=7) of array A091752 ((-1,2)Stirling2) divided by -12.

Original entry on oeis.org

1, 140, 27720, 7847840, 3049446400, 1564366003200, 1026108219136000, 838477001922560000, 835580445006827520000, 997744946185930342400000, 1406513375677181496524800000, 2311431202054422682730496000000, 4380418953582248141850148864000000, 9483101601549384660012281888768000000

Offset: 4

Wolfdieter Lang, Feb 27 2004

Cf. A052502, A091752.

a[n_] := (3/(4*5!))*n*(n-3)*(3*(n-2))!/((3^(n-2))*(n-2)!); Array[a, 15, 4] (* Amiram Eldar, Aug 30 2025 *)

a(n) = -A091752(n, 7)/12, n>=4.
a(n) = (3/(4*5!))*n*(n-3)*(3*(n-2))!/((3^(n-2))*(n-2)!) = 3*n*(n-3) * A052502(n-2)/(4*5!), n>=4.
G.f.: (x^4)*hypergeom([7/3, 8/3, 5, 2], [4], 9*x).
a(n) ~ 3^(2*n-7/2) * n^(2*n-2) /(160 * exp(2*n-5/n)). - Amiram Eldar, Aug 30 2025

A091756 Seventh column (m=8) of array A091752 ((-1,2)Stirling2).

Original entry on oeis.org

1, 220, 48720, 14463680, 5762556800, 3000655257600, 1987324218880000, 1634736979972096000, 1636859558116823040000, 1961447726093804748800000, 2772502956616965206835200000, 4565871212782705024303104000000, 8667353356325850744087642112000000, 18789301668434870837372923150336000000

Offset: 4

Wolfdieter Lang, Feb 27 2004

Cf. A052502, A091752.

a[n_] := (9/6!)*(n-3)*(9*n^3 - 45*n^2 + 36*n + 40)*(3*(n-3))!/((3^(n-3))*(n-3)!); Array[a, 15, 4] (* Amiram Eldar, Aug 30 2025 *)

a(n) = A091752(n, 8), n>=4.
a(n) = (9/6!)*(n-3)*(9*n^3-45*n^2+36*n+40)*(3*(n-3))!/((3^(n-3))*(n-3)!) = (9/6!)*(n-3)*(9*n^3-45*n^2+36*n+40) * A052502(n-3), n>=4.
a(n) ~ 3^(2*n-7/2) * n^(2*n-2) /(80 * exp(2*n-12/n)). - Amiram Eldar, Aug 30 2025

A052502 Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.

Original entry on oeis.org

1, 2, 40, 2240, 246400, 44844800, 12197785600, 4635158528000, 2345390215168000, 1524503639859200000, 1237896955565670400000, 1227993779921145036800000, 1461312598106162593792000000, 2054605512937264606871552000000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(3n) consisting of permutations whose cycle decomposition is a product of n disjoint 3-cycles.

F. W. J. Olver, Asymptotics and special functions, Academic Press, 1974, pages 336-344.

G. C. Greubel, Table of n, a(n) for n = 0..210
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 27
F. W. J. Olver et al., NIST Digital Library of Mathematical Functions, eq. 9.10.17.

Cf. A000142. Row sums of triangle A060063.
First column of array A091752 (also negative of second column).
Equals row sums of A157702. - Johannes W. Meijer, Mar 07 2009
Karol A. Penson suggested that the row sums of A060063 coincide with this entry.
Cf. A001147, A052504, A060706, A261317, A261381.
Trisection of column k=3 of A261430.

List([0..20], n-> Factorial(3*n)/(3^n*Factorial(n))) # G. C. Greubel, May 14 2019

[Factorial(3*n)/(3^n*Factorial(n)): n in [0..20]]; // G. C. Greubel, May 14 2019

spec := [S,{S=Set(Union(Cycle(Z,card=3)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[(3*n)!/(3^n*n!), {n, 0, 20}] (* G. C. Greubel, May 14 2019 *)

{a(n) = (3*n)!/(3^n*n!)}; \\ G. C. Greubel, May 14 2019

[factorial(3*n)/(3^n*factorial(n)) for n in (0..20)] # G. C. Greubel, May 14 2019

From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)
a(n) = (3*n)!/(3^n * n!).
a(n) ~ sqrt(3) * 9^n * (n/e)^(2n). (End)
E.g.f.: (every third coefficient of) exp(x^3/3).
G.f.: hypergeometric3F0([1/3, 2/3, 1], [], 9*x).
D-finite with recurrence a(n) = (3*n-1)*(3*n-2)*a(n-1) for n >= 1, with a(0) = 1.
Write the generating function for this sequence in the form A(x) = Sum_{n >= 0} a(n)* x^(2*n+1)/(2*n+1)!. The g.f. A(x) satisfies A'(x)*( 1 - A(x)^2) = 1. Robert Israel remarks that consequently A(x) is a root of z^3 - 3*z + 3*x with A(0) = 0. Cf. A001147, A052504 and A060706. - Peter Bala, Jan 02 2015
From Peter Bala, Feb 27 2024: (Start)
u(n) := a(n+1) satisfies the second-order recurrence u(n) = 18*n*u(n-1) + (3*n - 1)^2*(3*n - 2)^2*u(n-2) with u(0) = 2 and u(1) = 40.
A second solution to the recurrence is given by v(n) := u(n)*Sum_{k = 0..n} (-1)^k/((3*k + 1)*(3*k + 2)) with v(0) = 1 and v(1) = 18.
This leads to the continued fraction expansion (2/3)*log(2) = Sum_{k = 0..n} (-1)^k/((3*k + 1)*(3*k + 2)) = Limit_{n -> oo} v(n)/u(n) = 1/(2 + (1*2)^2/(18 + (4*5)^2/(2*18 + (7*8)^2/(3*18 + (10*11)^2/(4*18 +  ... ))))). (End)
From Gabriel B. Apolinario, Jul 30 2024: (Start)
a(n) = 3 * Integral_{t=0..oo} Ai(t)*t^(3*n) dt, where Ai(t) is the Airy function.
a(n) = Integral_{t=-oo..oo} Ai(t)*t^(3*n) dt. (End)

Edited by Wolfdieter Lang, Feb 13 2004

Title improved by Geoffrey Critzer, Aug 14 2015

A091535 First column (k=2) of array A091534 ((5,2)-Stirling2).

Original entry on oeis.org

1, 20, 1120, 123200, 22422400, 6098892800, 2317579264000, 1172695107584000, 762251819929600000, 618948477782835200000, 613996889960572518400000, 730656299053081296896000000, 1027302756468632303435776000000, 1684776520608556977634672640000000

Offset: 1

Wolfdieter Lang, Jan 23 2004

The scaled sequence (2/(3n-1)!!!)*a(n) = (3*n-2)!!! = A007559(n), n>=1.

Third column of array A091752 ((-1, 2)-Stirling2).

Cf. A007559, A032031, A091534.

a := n -> 9^n*GAMMA(n+1/3)*GAMMA(n+2/3)*sqrt(3)/(4*Pi);
seq(a(n), n=1..16); # Peter Luschny, Sep 17 2014

a[n_] := (3*n-1)!/(2!*3^(n-1)*(n-1)!); Array[a, 15] (* Amiram Eldar, Sep 01 2025 *)

a(n) = (3*n-1)!/(2!*3^(n-(2-1))*(n-1)!) = ((3*n-1)!/2)/A032031(n-1).
a(n) = A091534(n, 2), n>=1.
E.g.f.: (hypergeom([1/3, 2/3, 1], [], 9*x)-1)/2.
a(n) = 9^n*Gamma(n+1/3)*Gamma(n+2/3)*sqrt(3)/(4*Pi). - Peter Luschny, Sep 17 2014
a(n) ~ (sqrt(3)/2) * (3*n/e)^(2*n). - Amiram Eldar, Sep 01 2025

A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2n} T(n,k) x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.

Original entry on oeis.org

1, 0, 0, 1, 0, -2, 0, 0, 1, 2, 0, 0, -6, 0, 0, 1, 0, 0, 20, 0, 0, -12, 0, 0, 1, 0, -40, 0, 0, 80, 0, 0, -20, 0, 0, 1, 40, 0, 0, -360, 0, 0, 220, 0, 0, -30, 0, 0, 1, 0, 0, 1120, 0, 0, -1680, 0, 0, 490, 0, 0, -42, 0, 0, 1, 0, -2240, 0, 0, 9520, 0, 0, -5600, 0, 0, 952, 0, 0, -56, 0, 0, 1

Offset: 0

Werner Schulte, Jan 27 2020

Let r(s;n,x) = Sum_{k=0..s*n} A(s;n,k)*x^k = (-1)^n * e^(x^(s+1)/(s+1)) * (((d/dx)^n) e^(-x^(s+1)/(s+1))) for n >= 0 and x complex and some fixed integer s >= 1. Special cases: A(1;n,k) = A066325(n,k) and A(2;n,k) is this triangle. Formula: A(s;n,k) = (Sum_{i=0..floor(k/(s+1))} (-1)^i * binomial((n+k) /(s+1),i) * binomial(n+k-(s+1)*i,n)) * (-1)^(n-(n+k)/(s+1)) * (n!) / ((s+1)^((n+k)/(s+1)) * (((n+k)/(s+1))!)) if (n+k) mod (s+1) = 0 else 0 with n >= 0 and 0 <= k <= s*n.
  Recurrence: (1) A(s;n,k) = A(s;n-1,k-s) - (k+1) * A(s;n-1,k+1),
  (2) r(s;n,x) = x^s * r(s;n-1,x) - ((d/dx) r(s;n-1,x)) for n > 0 with initial values A(s;0,0) = 1 = r(s;0,x) and A(s;n,k) = 0 if k < 0 or k > s*n or (n+k) mod (s+1) > 0;
  E.g.f.: Sum_{n>=0} r(s;n,x)*t^n/(n!) = e^((x^(s+1)-(x-t)^(s+1))/(s+1)).
This generalization is result of a long and intensive discussion with Wolfdieter Lang. For more information see A091752.

			The irregular triangle T(n,k) starts:
n\k:  0     1    2    3    4     5   6     7   8   9  10   . . .      16
========================================================================
0  :  1
1  :  0     0    1
2  :  0    -2    0    0    1
3  :  2     0    0   -6    0     0   1
4  :  0     0   20    0    0   -12   0     0   1
5  :  0   -40    0    0   80     0   0   -20   0   0   1
6  : 40     0    0 -360    0     0 220     0   0 -30   0   0 1
7  :  0     0 1120    0    0 -1680   0     0 490   0   0 -42 0   0 1
8  :  0 -2240    0    0 9520     0   0 -5600   0   0 952   0 0 -56 0 0 1
etc.

Cf. A049404, A066325, A091752.

Row sums are (-1)^n*A252284(n).

T(n,k) = (-1)^k * (n!) * (Sum_{i=0..floor(k/3)} (-1)^i * binomial((n+k) /3,i) * binomial(n+k-3*i,n)) / (3^((n+k)/3) * ((n+k)/3)!) if (n+k) mod 3 = 0 else 0 with n >= 0 and 0 <= k <= 2*n.
Recurrence: (1) T(n,k) = T(n-1,k-2) - (k+1) * T(n-1,k+1),
  (2) T(n,k) = T(n-1,k-2) - 2*(n-1)*T(n-2,k-1) + (n-1)*(n-2)*T(n-3,k),
  (3) k*T(n,k) = 2*n*T(n-1,k-2) - n*(n-1)*T(n-2,k-1),
  (4) p(n,x) = x^2 * p(n-1,x) - (d/dx) p(n-1,x),
  (5) p(n,x) = x^2*p(n-1,x) - 2*(n-1)*x*p(n-2,x) + (n-1)*(n-2)*p(n-3,x),
  (6) (d/dx) p(n,x) = 2*n*x*p(n-1,x) - n*(n-1)*p(n-2,x) for n > 0 with initial values T(0,0) = 1 = p(0,x) and T(n,k) = 0 if k < 0 or k > 2*n or (n+k) mod 3 > 0.
T(n,2*n) = 1 for n >= 0.
T(3*n,0) = -T(3*n-1,1) = 2*T(3*n-2,2) = ((3*n)!)/(3^n * (n!)) for n > 0.
The polynomials p(n,x) satisfy for n >= 0 and x complex the differential equation: 0 = (((d/dx)^3) p(n,x)) - 2*x^2*(((d/dx)^2) p(n,x)) + (x^4 + 2*(n-1)*x) * ((d/dx) p(n,x)) - (2*n*x^3-(n+3)*n) * p(n,x).
E.g.f.: Sum_{n>=0} p(n,x)*t^n/(n!) = e^((x^3-(x-t)^3)/3).
((d/dx)^m) p(n,x) = Sum_{i=0..m} (-1)^i * binomial(m,i) * p(m-i,-x) * p(n+i,x) for m,n >= 0 and x complex.
T(3*n-k,k) = A091752(n+1,k+2) for 0 <= k <= 2*n.
(-1)^(n-k) * T(n,3*k-n) = A049404(n,k) for n > 0 and (n+2)/3 <= k <= n.

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A091753 Fourth column (m=5) of array A091752 ((-1,2)Stirling2) divided by -6.

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A091754 Fifth column (m=6) of array A091752 ((-1,2)Stirling2).

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A091755 Sixth column (m=7) of array A091752 ((-1,2)Stirling2) divided by -12.

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A091756 Seventh column (m=8) of array A091752 ((-1,2)Stirling2).

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A052502 Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.

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A091535 First column (k=2) of array A091534 ((5,2)-Stirling2).

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A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2*n} T(n,k) * x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.

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A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2n} T(n,k) x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.