A157284 Triangle T(n, k, m) = (m+1)^n*binomial(n,k)*f(n,m)*f(k,n-m)/n!, with T(n, 0, m) = 1, where f(n, k) = Product_{j=1..n} ( (1 - (k+1)^J)/(-k)^j ), f(n, 0) = n!, and m = 0, read by rows.

1, 1, 1, 1, 2, 4, 1, 3, 15, 105, 1, 4, 36, 744, 29016, 1, 5, 70, 3010, 389795, 121226245, 1, 6, 120, 9120, 2736000, 3065414400, 10017774259200, 1, 7, 189, 22995, 13452075, 37781497845, 471626437599135, 20185139902805378865, 1, 8, 280, 50960, 52234000, 308431323200, 10244546400088000, 1749976343076289328000, 1177042838234827583459440000

Offset

0,5

Links

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

J. Baik, T. Kriecherbauer, K. D. T.-R. McLaughlin, P. D. Miller, Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: announcement of results, International Mathematics Research Notices vol 2003, (2003) 821-858.

V. Gorin, Non-intersecting paths and Hahn orthogonal polynomial ensemble, arXiv:0708.2349 [math.PR], 2007.

Formula

T(n, k, m) = (m+1)^n*binomial(n,k)*f(n,m)*f(k,n-m)/n!, with T(n, 0, m) = 1, where f(n, k) = Product_{j=1..n} ( (1 - (k+1)^J)/(-k)^j ), f(n, 0) = n!, and m = 0.

Example

Triangle begins as:
  1;
  1, 1;
  1, 2,   4;
  1, 3,  15,   105;
  1, 4,  36,   744,    29016;
  1, 5,  70,  3010,   389795,   121226245;
  1, 6, 120,  9120,  2736000,  3065414400,  10017774259200;
  1, 7, 189, 22995, 13452075, 37781497845, 471626437599135, 20185139902805378865;

Mathematica

f[n_, k_]:= If[k==0, n!, QPochhammer[k+1, k+1, n]/(-k)^n];
T[n_, k_, m_]:= If[n==0, 1, (m+1)^n**Binomial[n, k]*f[n, m]*f[k, n-m]/n!];
Table[T[n, k, 0], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)

Prog

(Sage)
from sage.combinat.q_analogues import q_pochhammer
def f(n, k): return factorial(n) if (k==0) else q_pochhammer(n, k+1, k+1)/(-k)^n
def T(n, k, m): return 1 if (k==0) else (m+1)^n*binomial(n, k)*f(n, m)*f(k, n-m)/factorial(n)
[[T(n, k, 0) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 11 2022

Crossrefs

Cf. this sequence (m=0), A157285 (m=1).

Sequence in context: A162303 A128570 A371767 * A361523 A194733 A143973

Adjacent sequences: A157281 A157282 A157283 * A157285 A157286 A157287

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Jan 11 2022

Status

approved