A071951 -id:A071951 - cp's OEIS frontend

A089504 A generalization of triangle A071951 (Legendre-Stirling).

1, 6, 1, 36, 30, 1, 216, 756, 90, 1, 1296, 18360, 6156, 210, 1, 7776, 441936, 387720, 31356, 420, 1, 46656, 10614240, 23705136, 4150440, 119556, 756, 1, 279936, 254788416, 1432922400, 521757936, 29257200, 373572, 1260, 1, 1679616

Offset: 1

Wolfdieter Lang, Dec 01 2003

This triangle underlies the array entry A078741 ((3,3)-generalized Stirling2).

For the computation of the column sequences see A089505.

			[1]; [6,1]; [36,30,1]; [216,756,90,1]; ...
a(3,2) = 30 = ((-1)*(3*2*1)^1 + 4*(4*3*2)^1)/3.

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 8 rows.

Cf. A071951 (Legendre-Stirling, (2, 2) case).
The column sequences (without leading zeros) are A000400 (powers of 6), A089507, A089513-4, etc.
Cf. A008279, A089505, A089506.

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 2, 3]*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_{r=1..m} 1-fallfac(r+2, 3)*x with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m) = Sum_{p=1..m} A089505(m, p)*((p+2)*(p+1)*p)^(n-m))/D(m) if n>=m>=1 else 0; with D(m) := A089506(m).

A089278 Coefficient triangle for computation of column numbers of triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, -1, 3, 1, -15, 24, -7, 405, -2268, 2500, 2, -405, 6048, -20000, 16875, -11, 7425, -266112, 2000000, -4640625, 3176523, 143, -312741, 25474176, -390000000, 1879453125, -3344878719, 1927561216, -143, 995085, -178319232, 5250000000, -46986328125, 163899057231, -236126248960

Offset: 1

Wolfdieter Lang, Nov 07 2003

The k-th column sequence A071951(n+k,k), n>=0, is sum(a(k,p)*(p*(p+1))^n,p=1..k)/A089500(k), k>=1.

			[1]; [ -1,3]; [1,-15,24]; [ -7,405,-2268,2500]; ...
Sequence A071951(n+3,3)= A016309(n)= [1,20,292,...] has a(n)=
(1*(1*2)^n - 15*(2*3)^n + 24*(3*4)^n)/10.

W. Lang, First 7 rows.

a(n, m)= A089500(n)*(((-1)^(m+n))*(2*m+1)*((m*(m+1))^n)/((m+n+1)!*(n-m)!)).

A089500 Denominators for computation of column sequences of triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, 2, 10, 630, 2520, 277200, 97297200, 3405402000, 463134672000, 475176173472000, 16631166071520000, 4207685016094560000, 3786916514485104000000, 98459829376612704000000

Offset: 1

Wolfdieter Lang, Nov 07 2003

nonn

The k-th column sequence A071951(n+k,k), n>=0, is sum(A089278(k,p)*(p*(p+1))^n,p=1..k)/a(k), k>=1.

a(n)= N(n)/D(n) with N(n) := sfac(n-1)*sfac(2*n+1)/sfac(n+1)= A089501(n) and D(n) := gcd([seq((2*m+1)*((m*(m+1))^n)*N(n)/((n+m+1)!*(n-m)!), m=1..n)]), where sfac(n)=A000178(n) (superfactorials) and gcd(L) is the greatest common divisor >1 of a list of numbers L.

A071952 Diagonal T(n, 4) of triangle in A071951.

Original entry on oeis.org

1, 40, 1092, 25664, 561104, 11807616, 243248704, 4950550528, 100040447232, 2013177300992, 40412056994816, 810023815790592, 16221871691714560, 324694197936160768, 6496965245491888128, 129976281056339296256

Offset: 4

N. J. A. Sloane, Jun 16 2002

G. C. Greubel, Table of n, a(n) for n = 4..250
W. N. Everitt, L. L. Littlejohn and R. Wellman, Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148, 2002, 213-238.
L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181(2), 2002, 280-339.
Index entries for linear recurrences with constant coefficients, signature (40,-508,2304,-2880).

Cf. A000079, A016129, A015309, A089278, A089500.

Cf. A071951, A129467.

List([4..20], n-> 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315); # G. C. Greubel, Mar 16 2019

[2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315: n in [4..20]]; // G. C. Greubel, Mar 16 2019

Flatten[ Table[ Sum[(-1)^{r + 4}(2r + 1)(r^2 + r)^n/((r + 5)!(4 - r)!), {r, 1, 4}], {n, 4, 20}]]
LinearRecurrence[{40, -508, 2304, -2880}, {1, 40, 1092, 25664}, 20] (* G. C. Greubel, Mar 16 2019 *)

{a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315}; \\ G. C. Greubel, Mar 16 2019

[2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315 for n in (4..20)] # G. C. Greubel, Mar 16 2019

From Wolfdieter Lang, Nov 07 2003: (Start)
a(n+4) = A071951(n+4, 4) = (-7*2^n + 405*6^n - 2268*12^n + 2500*20^n)/630, n >= 0.
G.f.: x^4/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)). (End)
a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n-4), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467) and n > 3. - Mircea Merca, Apr 06 2013
From G. C. Greubel, Mar 16 2019: (Start)
a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315.
E.g.f.: (1 - exp(2*x))^4*(14 + 28*exp(2*x) + 28*exp(4*x) + 20*exp(6*x) + 10*exp(8*x) + 4*exp(10*x) + exp(12*x))/8!. (End)

More terms from Robert G. Wilson v, Jun 19 2002

Definition corrected by Georg Fischer, Jul 07 2025

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

A089274 Fifth column of the Legendre-Stirling triangle A071951.

Original entry on oeis.org

1, 70, 3192, 121424, 4203824, 137922336, 4380918784, 136378114048, 4191383868672, 127754693361152, 3873052857829376, 117001609550671872, 3526270158211870720, 106112798944292282368, 3189880933574260359168

Offset: 0

Wolfdieter Lang, Nov 07 2003

This is the fifth member of the family A000079 (powers of 2), A016129, A016309, A071952, etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (70,-1708,17544,-72000,86400).

Cf. A016129, A016309, A071951, A071952, A089278, A089500.

Cf. A000079 (powers of 2).

[(16875*(6*5)^n - 20000*(5*4)^n + 6048*(4*3)^n - 405*(3*2)^n + 2*(2*1)^n)/2520: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

Table[2^(n-3)*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)/315, {n,0,30}] (* G. C. Greubel, Nov 10 2024 *)

def A089274(n): return 2^n*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)//2520
[A089274(n) for n in range(31)] # G. C. Greubel, Nov 10 2024

G.f.: 1/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)*(1-6*5*x)).
a(n) = (16875*(6*5)^n -20000*(5*4)^n +6048*(4*3)^n -405*(3*2)^n +2*(2*1)^n)/2520.
a(n) = A071951(n+5, 5), n>=0.
a(n) = det(|ps(i+5,j+4)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). [Mircea Merca, Apr 06 2013]
E.g.f.: (1/2520)*(2*exp(2*x) - 405*exp(6*x) + 6048*exp(12*x) - 20000*exp(20*x) + 16875*exp(30*x)). - G. C. Greubel, Nov 10 2024

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).

A071953 Diagonal T(n,n-2) of triangle in A071951.

Original entry on oeis.org

4, 52, 292, 1092, 3192, 7896, 17304, 34584, 64284, 112684, 188188, 301756, 467376, 702576, 1028976, 1472880, 2065908, 2845668, 3856468, 5150068, 6786472, 8834760, 11373960, 14493960, 18296460, 22895964, 28420812, 35014252

Offset: 3

N. J. A. Sloane, Jun 16 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 3..5000
W. N. Everitt, L. L. Littlejohn and R. Wellman, Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148, 2002, 213-238.
L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181(2), 2002, 280-339.
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A071951, A071952.

List([3..30], n-> (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90); # G. C. Greubel, Mar 16 2019

[(n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90: n in [3..30]]; // G. C. Greubel, Mar 16 2019

Flatten[ Table[ Sum[(-1)^{r + n - 2}(2r + 1)(r^2 + r)^n/((r + n - 1)!(n - 2 - r)!), {r, 1, n - 2}], {n, 3, 34}]]
Table[(n-2)(n-1)n(n+1)(5n^2-11n+3)/90,{n,3,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{4,52,292,1092,3192, 7896,17304}, 30] (* Harvey P. Dale, Jul 03 2011 *)

{a(n) = (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90}; \\ G. C. Greubel, Mar 16 2019

[(n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90 for n in (3..30)] # G. C. Greubel, Mar 16 2019

a(n) = (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90.
a(0)=4, a(1)=52, a(2)=292, a(3)=1092, a(4)=3192, a(5)=7896, a(6)=17304, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jul 03 2011
G.f.: 4*(3*x*(x+2)+1)/(1-x)^7. - Harvey P. Dale, Jul 03 2011
E.g.f.: x^3*(60 + 135*x + 54*x^2 + 5*x^3)*exp(x)/90. - G. C. Greubel, Mar 16 2019

More terms from Robert G. Wilson v, Jun 19 2002

A089277 Sixth column of the triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, 112, 7896, 453056, 23232176, 1113673728, 51155215360, 2284897159168, 100157064553728, 4334351404617728, 185915811851773952, 7925465707325177856, 336395829865869340672, 14234737653310804590592

Offset: 0

Wolfdieter Lang, Nov 07 2003

Index entries for linear recurrences with constant coefficients, signature (112,-4648,89280,-808848,3110400,-3628800).

Cf. A071951, A071952, A089274, A089278, A089500, A129467.

G.f.: 1/Product_{p=1..6} (1 - p*(p+1)*x) = 1/((1-2*x)*(1-6*x)*(1-12*x)*(1-20*x)*(1-30*x)*(1-42*x)).
a(n) = A071951(n+6, 6), n>=0.
a(n) = (Sum_{p=1..6} A089278(6, p)*(p*(p+1))^n)/A089500(6) = (-11*2^n + 7425*6^n - 266112*12^n + 2000000*20^n - 4640625*30^n + 3176523*42^n)/277200.
a(n) = det(|ps(i+6,j+5)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013

A372343 a(n) is the permanent of the n X n matrix whose element (i,j) equals PS(i+2,j), where PS(r,c) is the Legendre-Stirling number of the second kind (A071951).

Original entry on oeis.org

1, 12, 2448, 2900160, 3335369728, 16355507060736, 202873109257748480, 5520786912662854893568, 304515605038514679874846720, 31568014831906551177163996921856, 5785425274398818300907155436515360768, 1783302045417843100606023721285336961122304, 886715046570481808433485979311322483302619676672

Offset: 0

Stefano Spezia, Apr 28 2024

nonn

G. E. Andrews, W. Gawronski, and L. L. Littlejohn, The Legendre-Stirling Numbers, Discrete Mathematics, Volume 311, Issue 14, 28 July 2011, Pages 1255-1272.

Cf. A069135 (determinant), A071951.

PS[n_,k_]:=Sum[(-1)^(r+k)(2r+1)(r^2+r)^n/((r+k+1)!(k-r)!),{r,0,k}]; a[0]:=1; a[n_]:=Permanent[Table[PS[i+2,j],{i,n},{j,n}]]; Array[a,13,0]

cp's OEIS Frontend

A089504 A generalization of triangle A071951 (Legendre-Stirling).

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A089278 Coefficient triangle for computation of column numbers of triangle A071951 (Legendre-Stirling).

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A089500 Denominators for computation of column sequences of triangle A071951 (Legendre-Stirling).

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A071952 Diagonal T(n, 4) of triangle in A071951.

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

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A089274 Fifth column of the Legendre-Stirling triangle A071951.

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

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A071953 Diagonal T(n,n-2) of triangle in A071951.

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