A157284 - OEIS

%I #20 Jun 24 2023 01:28:40

%S 1,1,1,1,2,4,1,3,15,105,1,4,36,744,29016,1,5,70,3010,389795,121226245,

%T 1,6,120,9120,2736000,3065414400,10017774259200,1,7,189,22995,

%U 13452075,37781497845,471626437599135,20185139902805378865,1,8,280,50960,52234000,308431323200,10244546400088000,1749976343076289328000,1177042838234827583459440000

%N 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.

%H G. C. Greubel, <a href="/A157284/b157284.txt">Rows n = 0..25 of the triangle, flattened</a>

%H J. Baik, T. Kriecherbauer, K. D. T.-R. McLaughlin, P. D. Miller, <a href="http://dx.doi.org/10.1155/S1073792803212125">Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: announcement of results</a>, International Mathematics Research Notices vol 2003, (2003) 821-858.

%H V. Gorin, <a href="http://arxiv.org/abs/0708.2349">Non-intersecting paths and Hahn orthogonal polynomial ensemble</a>, arXiv:0708.2349 [math.PR], 2007.

%F 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.

%e Triangle begins as:

%e   1;

%e   1, 1;

%e   1, 2,   4;

%e   1, 3,  15,   105;

%e   1, 4,  36,   744,    29016;

%e   1, 5,  70,  3010,   389795,   121226245;

%e   1, 6, 120,  9120,  2736000,  3065414400,  10017774259200;

%e   1, 7, 189, 22995, 13452075, 37781497845, 471626437599135, 20185139902805378865;

%t f[n_, k_]:= If[k==0, n!, QPochhammer[k+1, k+1, n]/(-k)^n];

%t T[n_, k_, m_]:= If[n==0, 1, (m+1)^n**Binomial[n,k]*f[n,m]*f[k,n-m]/n!];

%t Table[T[n,k,0], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 11 2022 *)

%o (Sage)

%o from sage.combinat.q_analogues import q_pochhammer

%o def f(n,k): return factorial(n) if (k==0) else q_pochhammer(n, k+1, k+1)/(-k)^n

%o 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)

%o [[T(n,k,0) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jan 11 2022

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

%K nonn,tabl,easy,less

%O 0,5

%A _Roger L. Bagula_, Feb 26 2009

%E Edited by _G. C. Greubel_, Jan 11 2022