A071951 -id:A071951 - cp's OEIS frontend

A016129 Expansion of 1/((1-2x)(1-6*x)).

1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296, 15116288, 90698752, 544194560, 3265171456, 19591036928, 117546237952, 705277460480, 4231664828416, 25389989101568, 152339934871552, 914039609753600, 5484237659570176, 32905425959518208, 197432555761303552

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Row sums of A100851.
Cf. A071951, A089278, A089500, A129467, A144843.
Sequences with gf 1/((1-n*x)*(1-6*x)): A000400 (n=0), A003464 (n=1), this sequence (n=2), A016137 (n=3), A016149 (n=4), A005062 (n=5), A053469 (n=6), A016169 (n=7), A016170 (n=8), A016172 (n=9), A016173 (n=10), A016174 (n=11), A016175 (n=12).

[(6^(n+1)-2^(n+1))/4 : n in [0..30]]; // Vincenzo Librandi, Oct 09 2011

Table[(6^(n+1) -2^(n+1))/4, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-12},{1,8},30] (* Harvey P. Dale, Jan 15 2015 *)

Vec(1/(1-2*x)/(1-6*x)+O(x^30)) \\ Charles R Greathouse IV, Apr 17 2012

[lucas_number1(n,8,12) for n in range(1, 31)] # Zerinvary Lajos, Apr 23 2009

[(6^n - 2^n)/4 for n in range(1,31)] # Zerinvary Lajos, Jun 04 2009

a(n) = A071951(n+2, 2) = 9*(2*3)^(n-1) - (2*1)^(n-1) = (2^(n-1))*(3^(n+1)-1), n>=0. - Wolfdieter Lang, Nov 07 2003
From Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005: (Start)
G.f.: 1/((1-2*x)*(1-6*x)).
E.g.f.: (-exp(2*x) + 3*exp(6*x))/2.
a(n) = (6^(n+1) - 2^(n+1))/4. (End)
a(n)^2 = A144843(n+1). - Philippe Deléham, Nov 26 2008
a(n) = 8*a(n-1) - 12*a(n-2). - Philippe Deléham, Jan 01 2009
a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013

A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

Original entry on oeis.org

1, 2, 4, 1, 4, 32, 38, 12, 1, 8, 208, 652, 576, 188, 24, 1, 16, 1280, 9080, 16944, 12052, 3840, 580, 40, 1, 32, 7744, 116656, 412800, 540080, 322848, 98292, 16000, 1390, 60, 1, 64, 46592, 1446368, 9196992, 20447056, 20453376, 10564304, 3047520, 511392, 50400

Offset: 1

N. J. A. Sloane, Dec 21 2002

A generalization of the Stirling2 numbers S_{1,1} from A008277.
The g.f. for column k=2*K is (x^K)*pe(K,x)*d(k,x) and for k=2*K+1 it is (x^K)*po(K,x)*2*(K+1)*K*d(k,x), K>= 1, with d(k,x) := 1/product(1-p*(p-1)*x,p=2..k) and the row polynomials pe(n,x) := sum(A089275(n,m)*x^m,m=0..n-1) and po(n,x) := sum(A089276(n,m)*x^m,m=0..n-1). - Wolfdieter Lang, Nov 07 2003
The formula for the k-th column sequence is given in A089511.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_2 (the disjoint union of n copies of the complete graph K_2). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k | 2    3    4    5    6   7   8
= = = = = = = = = = = = = = = = = =
  1 | 1
  2 | 2    4    1
  3 | 4   32   38   12    1
  4 | 8  208  652  576  188  24   1
...
Graph coloring interpretation of T(2,3) = 4: The graph 2K_2 is 2 copies of K_2, the complete graph on 2 vertices:
o---o  o---o
a   b  c   d
The four 3-colorings of 2K_2 are ac|b|d, ad|b|c, bc|a|d and bd|a|c. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
Steve Butler, Fan Chung, Jay Cummings, R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
Leonard Carlitz, On Arrays of Numbers, Am. J. Math., 54,4 (1932) 739-752. [Eqs. (3) and (4) with lambda = 0, mu = 2, a_{n,k-1} = a(n, k).- _Wolfdieter Lang_, Jan 30 2020 ]
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

Row sums give A020556. Triangle S_{1, 1} = A008277, S_{2, 1} = A008297 (ignoring signs), S_{3, 1} = A035342, S_{3, 2} = A078740, S_{3, 3} = A078741. A090214 (S_{4,4}).
The column sequences are A000079(n-1)(powers of 2), 4*A016129(n-2), A089271, 12*A089272, A089273, etc.
Main diagonal is A217900.
Cf. A071951 (Legendre-Stirling triangle).
Cf. A122193, A055203.

# Note that the function implements the full triangle because it can be
# much better reused and referenced in this form.
A078739 := proc(n,k) local r;
add((-1)^(n-r)*binomial(n,r)*combinat[stirling2](n+r,k),r=0..n) end:
# Displays the truncated triangle from the definition:
seq(print(seq(A078739(n,k),k=2..2*n)),n=1..6); # Peter Luschny, Mar 25 2011

t[n_, k_] := Sum[(-1)^(n-r)*Binomial[n, r]*StirlingS2[n+r, k], {r, 0, n}]; Table[t[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Apr 11 2013, after Peter Luschny *)

a(n, k) = sum(binomial(k-2+p, p)*A008279(2, p)*a(n-1, k-2+p), p=0..2) if 2 <= k <= 2*n for n>=1, a(1, 2)=1; else 0. Here A008279(2, p) gives the third row (k=2) of the augmented falling factorial triangle: [1, 2, 2] for p=0, 1, 2. From eq.(21) with r=2 of the Blasiak et al. paper.
a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*A008279(p, 2)^n, p=2..k) for 2 <= k <= 2*n, n>=1. From eq.(19) with r=2 of the Blasiak et al. paper.
a(n, k) = sum(A071951(n, j)*A089503(j, 2*j-k+1), j=ceiling(k/2)..min(n, k-1)), 1<=n, 2<=k<=2n; relation to Legendre-Stirling triangle. Wolfdieter Lang, Dec 01 2003
a(n, k) = A122193(n,k)*2^n/k! - Peter Luschny, Mar 25 2011
E^n = sum_{k=2}^(2n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^2d^2/dx^2.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*sum {k = 0..inf} (k*(k-1))^n*x^k/k!. - Peter Bala, Aug 15 2013

More terms from Wolfdieter Lang, Nov 07 2003

A016309 Expansion of 1/((1-2x)(1-6x)(1-12*x)).

Original entry on oeis.org

1, 20, 292, 3824, 47824, 585536, 7096384, 85576448, 1029436672, 12368356352, 148510974976, 1782675894272, 21395375902720, 256764101869568, 3081286768672768, 36976146501533696, 443717989683232768

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (20,-108,144)

Cf. A089278, A089500.

[(24*12^n-15*6^n+2^n)/10: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

CoefficientList[Series[1/((1-2x)(1-6x)(1-12x)),{x,0,30}],x] (* or *) LinearRecurrence[{20,-108,144},{1,20,292},30] (* Harvey P. Dale, Jul 26 2019 *)

a(n) = A071951(n+3, 3) = (24*12^n - 15*6^n + 2^n)/10. - Wolfdieter Lang, Nov 07 2003
a(n) = 18*a(n-1) - 72*a(n-2) + 2^n; a(n) = 20*a(n-1) - 108*a(n-2) + 144*a(n-3) for n > 2. - Vincenzo Librandi, Sep 02 2011
a(n) = det(|ps(i+3,j+2)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013

A084915 a(n) = (n!)^2*n.

Original entry on oeis.org

0, 1, 8, 108, 2304, 72000, 3110400, 177811200, 13005619200, 1185137049600, 131681894400000, 17526860144640000, 2753310393630720000, 504085244567224320000

Offset: 0

Jon Perry, Jul 14 2003

nonn

Used to prove that Sum_{n>=1} 1/A002378(n) = 1. Examining Sum_{n=1..k} 1/A002378(n) gives 1/2, 1/2 + 1/6, 1/2 + 1/6 + 1/12. Simplifying gives 1/2, 8/12, 108/144, where the numerators are this sequence and the denominators are A010790. Therefore we have k!^2*k/k!(k+1)! = k*k!/(k+1)! = k/(k+1), which tends to 1 as k tends to infinity.

			a(3) = 3!^2*3 = 36*3 = 108.

Cf. A002378, A010790, A071951.

PARI
```
for(n=1,50,print1(n!^2*n","))
```

a(n) = n!*(n+1)! - n!^2.

a(n) = det(PS(i+2,j+1), 1 <= i,j <= n-1), where PS(n,k) are Legendre-Stirling numbers of the second kind (A071951) and n > 0. [Mircea Merca, Apr 06 2013]

A089271 Third column (k=4) of array A078739(n,k) ((2,2)-generalized Stirling2).

Original entry on oeis.org

1, 38, 652, 9080, 116656, 1446368, 17636032, 213311360, 2569812736, 30898216448, 371141389312, 4455873443840, 53483541999616, 641880868118528, 7703040602324992, 92439308337643520, 1109288626710839296

Offset: 0

Wolfdieter Lang, Nov 07 2003

The numerator of the g.f. is the n=2 row polynomial of the triangle A089275.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Index entries for linear recurrences with constant coefficients, signature (20, -108, 144).

Cf. A089272, A071951 (Legendre-Stirling triangle).

[6*12^n-6*6^n+2^n: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

Table[6*12^n -6*6^n +2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{20,-108,144},{1,38,652},20] (* Harvey P. Dale, Oct 22 2024 *)

for(n=0,30, print1(6*12^n -6*6^n +2^n, ", ")) \\ G. C. Greubel, Feb 07 2018

G.f.: (1+18*x)/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)).

a(n) = 6*12^n - 6*6^n + 2^n = d(n) + 18*d(n-1), n>=1, a(0)=1, with d(n) := A016309(n) = A071951(n+3, 3) = (24*12^n-15*6^n+2^n)/10.

A089272 Fourth column (k=5) of array A078739(n,k) ((2,2)- generalized Stirling2) divided by 12.

Original entry on oeis.org

1, 48, 1412, 34400, 766416, 16296448, 337709632, 6896540160, 139644851456, 2813500878848, 56517475402752, 1133320271749120, 22702062218039296, 454469171469877248, 9094518828981174272, 181952003020274401280

Offset: 0

Wolfdieter Lang, Nov 07 2003

The numerator of the g.f. is the n=2 row polynomial of the triangle A089276.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003), 198-205.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Index entries for linear recurrences with constant coefficients, signature (40, -508, 2304, -2880).

Cf. A071952, A089271, A089273, A071951 (Legendre-Stirling triangle).

LinearRecurrence[{40, -508, 2304, -2880}, {1, 48, 1412, 34400}, 16] (* Jean-François Alcover, Feb 28 2020 *) (* Jean-François Alcover, Feb 28 2020 *)

G.f. (1+8*x)/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)).

a(n)= (3500*20^n - 3780*12^n + 945*6^n - 35*2^n)/630 = d(n) + 8*d(n-1), with d(n) := A071952(n+4)= (2500*20^n - 2268*12^n + 405*6^n - 7*2^n)/630, n>=1.

A130033 Fourth (m=3) column sequence of triangle A129467.

Original entry on oeis.org

1, -20, 508, -17544, 808848, -48405888, 3663035136, -342678781440, 38879803008000, -5263815891456000, 838682139211776000, -155393459730173952000, 33136711787903754240000, -8059211591488628981760000, 2217755736675770074398720000

Offset: 0

Wolfdieter Lang, May 04 2007

See the M. Bruschi et al. reference given in A129467.

			a(3)=-det([20,1,0],[292,40,1],[3824,1092,70])=-17544. [_Mircea Merca_, Apr 06 2013]

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

Cf. A010790 (m=1 column unsigned), A084915 (m=2 column unsigned).

Cf. A129467.

h:= func< n,k | (&+[1/j^k : j in [1..n]]) >;
A130033:= func< n | (-1)^n*(Factorial(n+2))^2*(2*(n+2) - (n+3)*h(n+2,2)) >;
[A130033(n): n in [0..30]]; // G. C. Greubel, Feb 10 2024

A130033[n_]:= (-1)^n*((n+2)!)^2*(2*(n+2) -(n+3)*HarmonicNumber[n+2,2]);
Table[A130033[n], {n,0,30}] (* G. C. Greubel, Feb 10 2024 *)

def A130033(n): return (-1)^n*(factorial(n+2))^2*(2*(n+2) - (n+3)*(zeta(2) - psi(1,n+3)))
[A130033(n) for n in range(31)] # G. C. Greubel, Feb 10 2024

a(n) = A129467(n+3,3),n>=0.
a(n) = (-1)^n*det(PS(i+3,j+2), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind (A071951). - Mircea Merca, Apr 06 2013
a(n) = (-1)^n * ((n+2)!)^2 * (2*(n+2) - (n+3)*h(n+2, 2)), where h(n,k) = Sum_{j=1..n} 1/j^k is the generalized harmonic number. - G. C. Greubel, Feb 10 2024

A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)x).

Original entry on oeis.org

1, 1, 3, 13, 81, 669, 6955, 88505, 1346209, 23998521, 493956467, 11596542533, 307301505073, 9110471500693, 299893197116059, 10888674034993905, 433549376981078593, 18833037527449398129, 888439543634687700579

Offset: 0

Paul D. Hanna, Dec 06 2007

nonn

			O.g.f.: A(x) = 1 + x/(1-2x) + x^2/((1-2x)*(1-6x)) + x^3/((1-2x)*(1-6x)*(1-12x)) + x^4/((1-2x)*(1-6x)*(1-12x)*(1-20x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,3,13,81,669,6955,88505,...] = BINOMIAL^2([1,1,5,31,253,2673,34833,..]);
[1,5,31,253,2673,34833,541879,...] = BINOMIAL^4([1,1,7,57,577,7389,...]);
[1,7,57,577,7389,115983,2151493,...] = BINOMIAL^6([1,1,9,91,1101,16497,...]);
[1,9,91,1101,16497,301669,..] = BINOMIAL^8([1,1,11,133,1873,32061,..]);
[1,11,133,1873,32061,666579,...] = BINOMIAL^10([1,1,13,183,2941,56529,...]);
etc.

Cf. A071951, A135920, A124373.

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-j*(j+1)*x+x*O(x^n))), n)

{a(n) = sum( k=0, n, sum( i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! ))} /* Michael Somos, Feb 25 2012 */

a(n+1) = row sums of A071951. - Michael Somos, Feb 25 2012

G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(k+1)*(k+2)*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

A089273 Fifth column (k=6) of array A078739(n,k) ((2,2)-generalized Stirling2).

Original entry on oeis.org

1, 188, 12052, 540080, 20447056, 706827968, 23178048832, 736079932160, 22912552596736, 704164858293248, 21462936995648512, 650674662791229440, 19656291799888777216, 592413643343696150528, 17826953303927872110592

Offset: 0

Wolfdieter Lang, Nov 07 2003

The numerator of the g.f. is the m=3 row polynomial of the triangle A089275.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003), 198-205.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Index entries for linear recurrences with constant coefficients, signature (70, -1708, 17544, -72000, 86400).

Cf. A089272, A071951 (Legendre-Stirling triangle).

a:= n-> (Matrix([[12052,188,1,0,0]]). Matrix(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [70,-1708,17544, -72000,86400][i] else 0 fi)^n)[1,3]: seq(a(n), n=0..30);  # Alois P. Heinz, Aug 14 2008

LinearRecurrence[{70, -1708, 17544, -72000, 86400}, {1, 188, 12052, 540080, 20447056}, 15] (* Jean-François Alcover, Feb 28 2020 *)

G.f.: (1+118*x+ 600*x^2)/Product_{p=1..5} (1-(p+1)*p*x).

a(n) = (2^n - 36*6^n + 36*6*12^n - 400*20^n + 75*3*30^n)/6 = d(n) + 118*d(n-1) + 600*d(n-2), n>=2, with d(n) := A089274(n)= A071951(n+5, 5)= (16875*30^n - 20000*20^n + 6048*12^n - 405*6^n + 2*2^n)/2520.

A089503 Triangle of numbers used for basis change between certain falling factorials.

Original entry on oeis.org

1, 1, 4, 1, 12, 30, 1, 24, 168, 336, 1, 40, 540, 2880, 5040, 1, 60, 1320, 13200, 59400, 95040, 1, 84, 2730, 43680, 360360, 1441440, 2162160, 1, 112, 5040, 117600, 1528800, 11007360, 40360320, 57657600, 1, 144, 8568, 274176, 5140800, 57576960

Offset: 1

Wolfdieter Lang, Dec 01 2003

Used to relate array A078739 ((2,2)-Stirling2) to triangle A071951 (Legendre-Stirling).

			The triangle begins:
n\m 1   2    3      4       5        6        7        8 ...
1:  1
2:  1   4
3:  1  12   30
4:  1  24  168    336
5:  1  40  540   2880    5040
6:  1  60 1320  13200   59400    95040
7:  1  84 2730  43680  360360  1441440  2162160
8:  1 112 5040 117600 1528800 11007360 40360320 57657600
...
Row 9:  1 144 8568 274176 5140800 57576960 374250240 1283143680 1764322560
Row 10: 1 180 13680 574560 14651280 234420480 2344204800 14065228800 45711993600 60949324800.
Reformatted - _Wolfdieter Lang_, Apr 10 2013
n=3: fallfac(x+2,6) = 1*fallfac(x,6) + 12*fallfac(x,5) + 30*fallfac(x,4).

Wolfdieter Lang, First 9 rows.

eq[n_, x_] := Sum[FactorialPower[x, 1 - m + 2*n]*a[n, m], {m, 1, n}] == FactorialPower[x + n - 1, 2*n]; eq[n_] := Table[eq[n, x], {x, n + 1, 2*n}]; row[n_] := First[Table[a[n, m], {m, 1, n}] /. Solve[eq[n]]]; Array[row, 10] // Flatten (* Jean-François Alcover, Sep 02 2016 *)
a[n_,m_]:= Binomial[n-1,m-1]*Binomial[2n,m-1]*Gamma[m]; Table[a[n,m],{n,1,10},{m,1,n}] (* Stefano Negro, Nov 10 2021 *)

fallfac(x+n-1, 2*n) = Sum_{m=1..n} a(n, m)*fallfac(x, 2*n-(m-1)), n>=1 where fallfac(x, k) := Product_{j=1..k} (x+1-j), with fallfac(n, k) = A068424(n, k) (falling factorials). a(n, m) = 0 if n < m.

T(n, m) = binomial(n-1, m-1)*binomial(2n, m-1)*m!, for 1 <= m <= n, with binomial(n, m) = A007318. - Stefano Negro, Nov 10 2021

cp's OEIS Frontend

A016129 Expansion of 1/((1-2*x)*(1-6*x)).

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A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

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A016309 Expansion of 1/((1-2*x)*(1-6*x)*(1-12*x)).

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A084915 a(n) = (n!)^2*n.

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A089271 Third column (k=4) of array A078739(n,k) ((2,2)-generalized Stirling2).

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PARI

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A089272 Fourth column (k=5) of array A078739(n,k) ((2,2)- generalized Stirling2) divided by 12.

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Mathematica

Formula

A130033 Fourth (m=3) column sequence of triangle A129467.

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Magma

A016129 Expansion of 1/((1-2x)(1-6*x)).

A016309 Expansion of 1/((1-2x)(1-6x)(1-12*x)).

A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)x).