A062139 -id:A062139 - cp's OEIS frontend

A090438 Generalized Stirling2 array (4,2).

1, 12, 8, 1, 360, 480, 180, 24, 1, 20160, 40320, 25200, 6720, 840, 48, 1, 1814400, 4838400, 4233600, 1693440, 352800, 40320, 2520, 80, 1, 239500800, 798336000, 898128000, 479001600, 139708800, 23950080, 2494800, 158400, 5940, 120, 1

Offset: 1

Wolfdieter Lang, Dec 23 2003

The row length sequences for this array is [1,3,5,7,9,11,...] = A005408(n-1), n>=1.

The scaled array a(n,k)/((2*n)!/k!) = A034870(n-1,k-2), n>=1, 2<=k<=2*n (Pascal triangle, even numbered rows only).

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
Wolfdieter Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A078740 (3, 2)-Stirling2.
Cf. A072678 (row sums), A090439 (alternating row sums).
Cf. A062139.

with(PolynomialTools):
p := n -> (2*n+2)!*hypergeom([-2*n],[3], -x)/2:
seq(CoefficientList(simplify(p(n)),x), n=0..5); # Peter Luschny, Apr 08 2015

a[n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*Product[FactorialPower[p + 2*(j-1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2 n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

Recursion: a(n, k) = sum(binomial(2, p)*fallfac(2*(n-1)-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=4, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+2*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=4, s=2.
a(n, k) = ((2*n)!/k!)*binomial(2*(n-1), k-2), n>=1, 2<=k<=2*n.
E.g.f. column k>=2 (with leading zeros): (((-1)^k)/k!)*(sum(((-1)^p)*binomial(k, p)*hypergeom([(p-1)/2, p/2], [], 4*x), p=2..k)-(k-1)).
Coefficient triangle of the polynomials (2*n+2)!*hypergeom([-2*n],[3],-x)/2. - Peter Luschny, Apr 08 2015
Coefficient triangle of Laguerre polynomials (2*n)!*L(2*n,2,-x). - Peter Luschny, Apr 08 2015

A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).

Original entry on oeis.org

1, 6, -1, 42, -14, 1, 336, -168, 24, -1, 3024, -2016, 432, -36, 1, 30240, -25200, 7200, -900, 50, -1, 332640, -332640, 118800, -19800, 1650, -66, 1, 3991680, -4656960, 1995840, -415800, 46200, -2772, 84, -1, 51891840, -69189120

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,5,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^6. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials sum(|A008297(n,m)|*(-x)^m, m=1..n), n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
These polynomials appear in the radial part of the l=2 (d-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148-A062152. Row sums (signed) give A062191; row sums (unsigned) give A062192.
The unsigned version of this triangle is the triangle of unsigned 3-Lah numbers A143498. - Peter Bala, Aug 25 2008

			Triangle begins:
  {1};
  {6, -1};
  {42, -14, 1};
  {336, -168, 24, -1};
  ...
2!*L(2, 5, x) = 42-14*x+x^2.

A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.

Indranil Ghosh, Rows 0..125, flattened
Index entries for sequences related to Laguerre polynomials

Cf. A021009, A062137, A062139, A062140, A066667.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148, A062149, A062150, A062151, A062152.
Row sums (signed) give A062191, row sums (unsigned) give A062192.
Cf. A143498.

Flatten[Table[((-1)^m)*n!*Binomial[n+5,n-m]/m!,{n,0,8},{m,0,n}]] (* Indranil Ghosh, Feb 24 2017 *)

tabl(nn) = {for (n=0, nn, for (m=0, n, print1(((-1)^m)*n!*binomial(n+5, n-m)/m!, ", "); ); print(); ); } \\ Indranil Ghosh, Feb 24 2017

row(n) = Vecrev(n!*pollaguerre(n, 5)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n, r):return f(n)//f(r)//f(n-r)
i=-1
for n in range(26):
    for m in range(n+1):
        i += 1
        print(str(i)+" "+str(((-1)**m)*f(n)*C(n+5, n-m)//f(m))) # Indranil Ghosh, Feb 24 2017

T(n, m) = ((-1)^m)*n!*binomial(n+5, n-m)/m!.

E.g.f. for m-th column: ((-x/(1-x))^m)/(m!*(1-x)^6), m >= 0.

A235706 (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 6, 18, 9, 1, 0, 24, 36, 12, 1, 0, 0, 60, 60, 15, 1, 0, 0, 0, 120, 90, 18, 1, 0, 0, 0, 0, 210, 126, 21, 1, 0, 0, 0, 0, 0, 336, 168, 24, 1, 0, 0, 0, 0, 0, 0, 504, 216, 27, 1, 0, 0, 0, 0, 0, 0, 0, 720, 270, 30, 1

Offset: 0

Tom Copeland, Apr 20 2014

The associated Laguerre polynomials n!*Lag(n,3-n,-x) are related to the rook polynomials of a rectangular 3 X n chessboard by R(3,n,x) = n!*x^n*Lag(n,3-n,-1/x), which are also the matching polynomials, or generating function of the number of k-edge matchings, of the complete bipartite graph K(m,n), or biclique (cf. Wikipedia for details).

The formulas here and below can be naturally extended with 3 replaced by any positive integer m. For m = 1 and 2, see unsigned A132013 and A132014. The formulas there can be extrapolated to apply to this matrix.

			Triangle begins:
  1;
  3,  1;
  6,  6,  1;
  6, 18,  9,  1;
  0, 24, 36, 12,  1;
  0,  0, 60, 60, 15, 1;
  ...

Michel Marcus, Rows n = 0..50 of triangle, flattened

Cf. A007318, A008306 for a generalization, A132013, A132014, A132440, A238363, A238385.
....................................
With 0th row: 1
n-th row: n!*Lag(n,3-n,-x)
....................................
1st: 1!*Lag(1,2,-x)  = A062139(1,k,-x)
2nd: 2!*Lag(2,1,-x)  = A105278(2,k,x)
3rd: 3!*Lag(3,0,-x)  = A021009(3,k,-x)
4th: 4!*Lag(4,-1,-x) = A111596(4,k,-x)
5th: 5!*Lag(5,-2,-x) = cf. x^2*A062139(3,k,x)
6th: 6!*Lag(6,-3,-x) = cf. x^3*A062137(3,k,-x)
....................................
n-th row: x^(n-3)*3!*Lag(3,n-3,-x)
....................................
1st: x^(-2)*3!Lag(3,-2,-x) = cf. x^(-2)*[x^2*A062139(1,k,x)]
2nd: x^(-1)*3!Lag(3,-1,-x) = x^(-1)*A111596(3,k,-x)
3rd: x^0*3!Lag(3,0,-x)     = x^0*A021009(3,k,-x)
4th: x^1*3!Lag(3,1,-x)     = x^1*A105278(3,k,x)
5th: x^2*3!Lag(3,2,-x)     = x^2*A062139(3,k,-x)
6th: x^3*3!Lag(3,3,-x)     = x^3*A062137(3,k,-x)

/* As triangle */ [[Binomial(3, n-k)*Factorial(n)/Factorial(k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 28 2017

Table[Binomial[3, n - k] n! / k!, {n, 0, 9}, {k, 0, n}]//Flatten (* Vincenzo Librandi, Jul 28 2017 *)

T(n,k) = binomial(3,n-k)*n!/k!
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 28 2017

row(n) = Vecrev(n!*pollaguerre(n, 3-n, -x)); \\ Michel Marcus, Feb 06 2021

T(n,k) = binomial(3,n-k)*n!/k! = binomial(n,k)*3!/(3-n+k)!.
E.g.f.: exp(y*x)(1+y)^3, so this is an Appell sequence of polynomials with lowering operator L= D= d/dx and raising operator R = x + 3/(1+D).
E.g.f. of inverse matrix is exp(x*y)/(1+y)^3.
Multiply the n-th diagonal of the Pascal matrix A007318 by d(0)=1, d(1)=3, d(2)=6, d(3)=6, and d(n)=0 for n>3 to obtain T.
Row polynomials: n!*Lag(n,3-n,-x) = x^(n-3)*3!*Lag(3,n-3,-x) =
(3!/(3-n)!)*K(-n,3-n+1,-x) where K is Kummer's confluent hypergeometric function (as a limit of n+c as c tends to zero).
T = (I + A132440)^3 = exp[3*(A238385-I)]. I = identity matrix.
Operationally, n!Lag(n,3-n,-:xD:) = x^(n-3)*:Dx:^n*x^(3-n) = x^(-3)*:xD:^n*x^3 = n!*binomial(xD+3,n) = n!*binomial(3,n)*K(-n,3-n+1,-:xD:) where :AB:^n = A^n*B^n for any two operators.
n-th row polynomial: n!*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*Lag(k,3,-x). - Peter Bala, Jul 25 2021

A355372 Expansion of the e.g.f. log((1 - x) / (1 - 2*x)) / (1 - x)^3.

Original entry on oeis.org

0, 1, 9, 77, 714, 7374, 85272, 1102968, 15908400, 254866320, 4516084800, 88102382400, 1883199024000, 43885950595200, 1109416142822400, 30273281955302400, 887493144729139200, 27827941161784780800, 929449073791558656000, 32943696020637889536000, 1234946945823695419392000

Offset: 0

Mélika Tebni, Jun 30 2022

nonn

Conjecture: For p prime, a(p) == -1 (mod p).

Cf. A000292, A062139, A355171.

A355372 := n -> A000292(n)*n!*hypergeom([1 - n, 1, 1], [2, 4], -1):
seq(simplify(A355372(n)), n = 0..20);

CoefficientList[Series[Log[(1 - x)/(1 - 2*x)]/ (1 - x)^3,{x,0,20}],x]Table[n!,{n,0,20}] (* Stefano Spezia, Jun 30 2022 *)

a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A062139(n, k + 1).
a(0) = 0, a(n) = n!*Sum_{k=1..n} (n-k+2)*(n-k+1)*(2^k-1)/(2*k).
a(n) = A000292(n)*n!*hypergeom([1 - n, 1, 1], [2, 4], -1). - Peter Luschny, Jun 30 2022

A343861 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n,x) (rising powers of x).

Original entry on oeis.org

1, 2, -1, 12, -8, 1, 120, -90, 18, -1, 1680, -1344, 336, -32, 1, 30240, -25200, 7200, -900, 50, -1, 665280, -570240, 178200, -26400, 1980, -72, 1, 17297280, -15135120, 5045040, -840840, 76440, -3822, 98, -1, 518918400, -461260800, 161441280, -29352960, 3057600, -188160, 6720, -128, 1

Offset: 0

Seiichi Manyama, May 01 2021

			The triangle begins:
       1;
       2,      -1;
      12,      -8,      1;
     120,     -90,     18,     -1;
    1680,   -1344,    336,    -32,    1;
   30240,  -25200,   7200,   -900,   50,  -1;
  665280, -570240, 178200, -26400, 1980, -72, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

For k=0..1 the (unsigned) columns give A001813, A092956(n-1).
Row sums (signed) give A006902, row sums (unsigned) give A082545.
Cf. A066667, A062137, A062138, A062139, A062140.

[(-1)^k*Factorial(n-k)*Binomial(n,k)*Binomial(2*n, n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 11 2022

T[n_, k_] := (-1)^k * (2*n)! * Binomial[n, k]/(k + n)!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

T(n, k) = (-1)^k*(2*n)!*binomial(n,k)/(k+n)!;

PARI
```
row(n) = Vecrev(n!*pollaguerre(n, n));
```

def A343861(n,k): return (-1)^k*factorial(n-k)*binomial(n,k)*binomial(2*n,n+k)
flatten([[A343861(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 11 2022

T(n, k) = (-1)^k * n! * binomial(2*n,n-k)/k! = (-1)^k * (2*n)! * binomial(n,k)/(k+n)!.
T(n, 0) = A001813(n).
T(n, 1) = -A092956(n-1).
Sum_{k=0..n} T(n, k) = A006902(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A082545(n).

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A090438 Generalized Stirling2 array (4,2).

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A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).

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A235706 (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).

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A355372 Expansion of the e.g.f. log((1 - x) / (1 - 2*x)) / (1 - x)^3.

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A343861 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n,x) (rising powers of x).

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