A089504 -id:A089504 - cp's OEIS frontend

A089505 Triangle of signed numbers used for the computation of the column sequences of triangle A089504.

1, -1, 4, 1, -24, 50, -1, 114, -950, 1350, 31, -15504, 400520, -1897200, 2052855, -9269, 19612560, -1431859000, 17333030000, -56265334125, 49236404224, 342953, -3011508588, 594221485000, -16634292228125, 123422029355625, -302409994743808, 222337901418633, -9945637

Offset: 1

Wolfdieter Lang, Dec 01 2003

A089504(n+m,m)= sum(a(m,p)*((p+2)*(p+1)*p)^n,p=1..m)/D(m) with D(m) := A089506(m); m=1,2,..., n>=0.

			[1]; [ -1,4]; [1,-24,50]; [ -1,114,-950,1350]; ...
a(3,2)= -24 = 27*(-1)*((4*3*2)^2)/((4*3*2-3*2*1)*(5*4*3-4*3*2)).
A089504(2+3,3) = A089513(2) = 6156 = (1*(3*2*1)^2 - 24*(4*3*2)^2 + 50*(5*4*3)^2)/27.

W. Lang, First 7 rows.

Companion denominator sequence is A089506.

b[n_, m_] := (-1)^(n - m)*FactorialPower[m + 2, 3]^(n - 1)/(Product[ FactorialPower[m + 2, 3] - FactorialPower[r + 2, 3], {r, 1, m - 1}] * Product[ FactorialPower[r + 2, 3] - FactorialPower[m + 2, 3], {r, m + 1, n}]); den[n_] := LCM @@ Table[ Denominator[b[n, m]], {m, 1, n}]; a[n_, m_] := den[n]*b[n, m]; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 02 2016 *)

a(n, m)= D(n)*((-1)^(n-m))*(((m+2)*(m+1)*m)^(n-1))/(product(fallfac(m+2, 3)-fallfac(r+2, 3), r=1..m-1)*product(fallfac(r+2, 3)-fallfac(m+2, 3), r=m+1..n)), with D(n) := A089506(n) and fallfac(n, m) := A008279(n, m) (falling factorials), 1<=m<=n else 0. (Replace in the denominator the first product by 1 if m=1 and the second one by 1 if m=n.)

a(n, m)= A089506(n)*((-1)^(n-m))*(fallfac(m+2, 3)^(n-1))*(3*m^2+6*m+2)/((n-m)!*(m-1)!*product(fallfac(m+r+2, 2)-r*m, r=1..n)), n>=m>=1.

A089513 Third column of triangle A089504.

Original entry on oeis.org

1, 90, 6156, 387720, 23705136, 1432922400, 86230132416, 5179923146880, 310942155338496, 18660051727004160, 1119687641441381376, 67183287394552842240, 4031045937469026349056, 241863924899255181189120

Offset: 0

Wolfdieter Lang, Dec 01 2003

Convolution of A089507 with powers of 60.

Harvey P. Dale, Table of n, a(n) for n = 0..562
Index entries for linear recurrences with constant coefficients, signature (90,-1944,8640).

Cf. A089505, A089507, A089514.

CoefficientList[Series[1/(1-90 x+1944 x^2-8640 x^3),{x,0,20}],x] (* or *) LinearRecurrence[{90,-1944,8640},{1,90,6156},20] (* Harvey P. Dale, Jul 07 2021 *)

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)).
a(n)=A089504(n+3, 3), n>=0.
a(n)= (50*(5*4*3)^n - 24*(4*3*2)^n + (3*2*1)^n)/27 = (6^n)*(5*10^(n+1) - 3*2^(2*n+3) + 1)/27.

A090215 A generalization of triangles A071951 (Legendre-Stirling) and A089504.

Original entry on oeis.org

1, 24, 1, 576, 144, 1, 13824, 17856, 504, 1, 331776, 2156544, 199296, 1344, 1, 7962624, 259117056, 73903104, 1328256, 3024, 1, 191102976, 31102009344, 26864234496, 1189638144, 6408576, 6048, 1, 4586471424, 3732432224256, 9702226427904, 1026160275456, 11956045824, 24697728, 11088, 1

Offset: 1

Wolfdieter Lang, Dec 01 2003

This triangle underlies the array entry A090214 ((4,4)-generalized Stirling2).

			[1]; [24,1]; [576,144,1]; [13824,17856,504,1]; ...

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
Wolfdieter Lang, First 8 rows.

Cf. A071951 (Legendre-Stirling, (2, 2) case), A089504 ((3, 3)-case).

The column sequences (without leading zeros) are A009968 (powers of 24), etc.

max = 10; f[m_] := 1/Product[1-FactorialPower[r+3, 4]*x, {r, 1, m}]; col[m_] := CoefficientList[f[m] + O[x]^(max-m+1), x]; a[n_, m_] := col[m][[n-m+1]]; Table[a[n, m], {n, 1, max}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

G.f. for m-th column sequence (without leading zeros and m>=1) is 1/product(1-fallfac(r+3, 4)*x, r=1..m) with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m) = sum(A089515(m, p)*fallfac(p, 4)^(n-m), p=1..m)/D(m) if n>=m>=1 else 0; with D(m) := A089516(m).

More terms coming from a-file added by Michel Marcus, Feb 08 2023

A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

Original entry on oeis.org

1, 30, 756, 18360, 441936, 10614240, 254788416, 6115201920, 146766525696, 3522406694400, 84537821131776, 2028908069959680, 48693795855814656, 1168651113600245760, 28047626804770062336, 673143043784666480640

Offset: 0

Wolfdieter Lang, Dec 01 2003

Convolution of A000400 (powers of 6) with A009968 (powers of 24).

Vincenzo Librandi, Table of n, a(n) for n = 0..730
Index entries for linear recurrences with constant coefficients, signature (30,-144).

Cf. A089505, A089513, A089514.

[6^n*(4^(n+1)-1)/3: n in [0..15]]; // Vincenzo Librandi, Oct 18 2017

CoefficientList[Series[1/((1-6x)(1-24x)),{x,0,20}],x] (* or *) LinearRecurrence[{30,-144},{1,30},20] (* Harvey P. Dale, Sep 25 2017 *)

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)).
a(n) = A089504(n+2, 2), n>=0.
a(n) = (4*(4*3*2)^n - (3*2*1)^n)/3 = (2^n)*(2^(2*(n+1))-1)*3^(n-1).
a(n) = 6^n*(4^(n+1)-1)/3. - Vincenzo Librandi, Oct 18 2017

A089514 Fourth column of triangle A089504.

Original entry on oeis.org

1, 210, 31356, 4150440, 521757936, 64043874720, 7771495098816, 937759335004800, 112842062355914496, 13559707534436743680, 1628284591773850622976, 195461334300256627599360

Offset: 0

Wolfdieter Lang, Dec 01 2003

Convolution of A089513 with powers of 120.

Harvey P. Dale, Table of n, a(n) for n = 0..480
Index entries for linear recurrences with constant coefficients, signature (210,-12744,241920,-1036800).

Cf. A089505, A089513.

CoefficientList[Series[1/((1-6x)(1-24x)(1-60x)(1-120x)),{x,0,20}],x] (* or *) LinearRecurrence[ {210,-12744,241920,-1036800},{1,210,31356,4150440},20] (* Harvey P. Dale, Mar 17 2023 *)

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)*(1-6*5*4*x)).
a(n) = A089504(n+4, 4), n>=0.
a(n) = (1350*(6*5*4)^n - 950*(5*4*3)^n + 114*(4*3*2)^n - (3*2*1)^n)/513.

A089506 Denominators used in A089505 to compute column sequences of triangle A089504.

Original entry on oeis.org

1, 3, 27, 513, 540702, 8891844390, 27306854121690, 94235953573952190, 1684561906087969348440, 1106757172299795861925080, 8644064410182787098480243878401800, 289900692457541891469406183557646377576408600, 1206267931807293851480420134690207296382114287151076400

Offset: 1

Wolfdieter Lang, Dec 01 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..50

a[n_, m_] := (-1)^(n-m)*FactorialPower[m+2, 3]^(n-1)/(Product[ FactorialPower[m+2, 3] - FactorialPower[r+2, 3], {r, 1, m-1}]*Product[ FactorialPower[r+2, 3] - FactorialPower[m+2, 3], {r, m+1, n}]); a[n_] := LCM @@ Table[Denominator[a[n, m]], {m, 1, n}]; Array[a, 11] (* Jean-François Alcover, Sep 02 2016 *)

a(n) = lcm(seq(denominator(a(n, m)), m=1..n)) with the a(n, m) formula given in A089505(n, m) but without the D(n) factor in front and lcm denotes the least common multiple of a set of numbers.

a(12)-a(13) from Vincenzo Librandi, Mar 15 2018

A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

Original entry on oeis.org

1, 6, 18, 9, 1, 36, 540, 1242, 882, 243, 27, 1, 216, 13608, 94284, 186876, 149580, 56808, 11025, 1107, 54, 1, 1296, 330480, 6148872, 28245672, 49658508, 41392620, 18428400, 4691412, 706833, 63375, 3285, 90, 1, 7776, 7954848, 380841264, 3762380016, 13062960720

Offset: 1

N. J. A. Sloane, Dec 21 2002

The sequence of row lengths for this array is [1,4,7,10,..]= A016777(n-1), n>=1.
The g.f. for the k-th column, (with leading zeros and k>=3) is G(k,x)= x^ceiling(k/3)*P(k,x)/product(1-fallfac(p,3)*x,p=3..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A089517(k,m)*x^m,m=0..kmax(k)), k>=3, with kmax(k) := A004523(k-3)= floor(2*(k-3)/3)= [0,0,1,2,2,3,4,4,5,...]. For the recurrence of the G(k,x) see A089517. Wolfdieter Lang, Dec 01 2003
For the computation of the k-th column sequence see A090219.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_3 (the disjoint union of n copies of the complete graph K_3). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k |   3     4     5      6      7     8     9   10  11  12
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
  1 |   1
  2 |   6    18     9      1
  3 |  36   540  1242    882    243    27     1
  4 | 216 13608 94284 186876 149580 56808 11025 1107  54   1
...
Graph coloring interpretation of T(2,3) = 6:
The graph 2K_3 is 2 copies of K_3, the complete graph on 3 vertices:
    o b      o e
   / \      / \
  o---o    o---o
  a   c    d   f
The six 3-colorings of 2K_3 are ad|be|cf, ad|bf|ce, ae|bd|cf, ae|bf|cd, af|bd|ce, and af|be|cd. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
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.
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.
W. Lang, First 6 rows.

Row sums give A069223. Cf. A078739.
The column sequences (without leading zeros) are A000400 (powers of 6), 18*A089507, 9*A089518, A089519, etc.
A089504, A069223 (row sums), A090212 (alternating row sums).
A078740, A090214.

a[n_, k_] := (-1)^k*Sum[(-1)^p*((p-2)*(p-1)*p)^n*Binomial[k, p], {p, 3, k}]/k!; Table[a[n, k], {n, 1, 6}, {k, 3, 3*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

a(n, k) = (((-1)^k)/k!)*Sum_{p = 3..k} (-1)^p* binomial(k, p)*fallfac(p, 3)^n, with fallfac(p, 3) := A008279(p, 3) = p*(p-1)*(p-2); 3 <= k <= 3*n, n >= 1, else 0. From eq.(19) with r = 3 of the Blasiak et al. reference.
E^n = Sum_{k = 3..3*n} a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^3d^3/dx^3.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k >= 0} (k*(k-1)*(k-2))^n*x^k/k!. - Peter Bala, Aug 15 2013

A090217 A generalization of triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, 120, 1, 14400, 840, 1, 1728000, 619200, 3360, 1, 207360000, 447552000, 9086400, 10080, 1, 24883200000, 322444800000, 23345280000, 76824000, 25200, 1, 2985984000000, 232185139200000, 59152550400000, 539602560000, 457848000

Offset: 1

Wolfdieter Lang, Dec 01 2003

This is the fourth member of the family A071951 (Legendre-Stirling,(2,2) case), A089504((3,3)-case), A090215 ((4,4)-case).

This triangle underlies the array entry A090216 ((5,5)-generalized Stirling2).

			Triangle starts:
[1];
[120,1];
[14400,840,1];
[1728000,619200,3360,1];
...

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
W. Lang, First 5 rows.

The column sequences (without leading zeros) are powers of 120, etc.

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 4, 5]*x, {r, 1, m}]; col[m_] := CoefficientList[f[m] + O[x]^(max - m + 1), x]; a[n_, m_] := col[m][[n - m + 1]]; Table[a[n, m], {n, 1, max}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 02 2016 *)

G.f. for m-th column (without leading zeros and m>=1) is 1/product(1-fallfac(r+4, 5)*x, r=1..m) with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m)=sum(A090435(m, p)*fallfac(p, 5)^(n-m), p=1..m)/D(m) if n>=m>=1 else 0; with D(m) := A090436(m).

A089518 Third column (k=5) of array A078741 ((3,3)-Stirling2) divided by 9.

Original entry on oeis.org

1, 138, 10476, 683208, 42315696, 2570768928, 155010407616, 9318969502848, 559578466388736, 33585275183251968, 2015370124337581056, 120928294183739148288, 7255843732407562776576, 435354129897768445943808

Offset: 0

Wolfdieter Lang, Dec 01 2003

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.
Index entries for linear recurrences with constant coefficients, signature (90, -1944, 8640).

Cf. A089513 (third column of A089504), A089519, A090219.

G.f.: (1+48*x)/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)).

a(n)= (10*(5*4*3)^n - 8*(4*3*2)^n + (3*2*1)^n)/3 = b(n) + 48*b(n-1), with b(n) := A089513(n).

cp's OEIS Frontend

A089505 Triangle of signed numbers used for the computation of the column sequences of triangle A089504.

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A089513 Third column of triangle A089504.

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A090215 A generalization of triangles A071951 (Legendre-Stirling) and A089504.

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A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

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A089514 Fourth column of triangle A089504.

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A089506 Denominators used in A089505 to compute column sequences of triangle A089504.

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A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

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A090217 A generalization of triangle A071951 (Legendre-Stirling).

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