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%I #23 Jun 24 2023 01:26:56
%S 1,2,2,6,12,18,28,84,336,1456,210,840,6300,88200,1874250,2604,13020,
%T 156240,4843440,377788320,59010535584,54684,328104,5741820,329197680,
%U 63946649340,39774815889480,61856467844122980,1984248,13889736
%N Triangle T(n, k, m) = (m+1)^n*t(n, m)*t(k, n-m)/(k! * (n-k)!), where T(0, k, m) = 1, t(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (m+1)^i ), and t(n, 0) = n!, read by rows.
%H G. C. Greubel, <a href="/A157285/b157285.txt">Rows n = 0..30 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 preprint arXiv:0708.2349 [math.PR], 2007.
%F T(n, k, m) = (m+1)^n*t(n, m)*t(k, n-m)/(k! * (n-k)!), where T(0, k, m) = 1, t(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (m+1)^i ), and t(n, 0) = n!.
%F T(n, k, m) = (1/n!)*binomial(n, k)*(m+1)^n*t(n, m)*t(k, n-m), with T(1, k, m) = 2, and t(n, k) = (1/m^n)*Product_{j=1..n} ((m+1)^j - 1). - _G. C. Greubel_, Jul 09 2021
%e Triangle begins as:
%e 1;
%e 2, 2;
%e 6, 12, 18;
%e 28, 84, 336, 1456;
%e 210, 840, 6300, 88200, 1874250;
%e 2604, 13020, 156240, 4843440, 377788320, 59010535584;
%t (* First program *)
%t t[n_, k_] = If[k==0, n!, Product[Sum[(k+1)^i, {i, 0, j-1}], {j, n}]];
%t T[n_, k_, m_] = If[n==0, 1, ((m+1)^n*t[n, m]*t[k, n-m])/(k!*(n-k)!)];
%t Flatten@Table[T[n, k, 1], {n,0,10}, {k,0,n}]
%t (* Second program *)
%t t[n_, m_] = (1/m^n)*Product[(m+1)^j - 1, {j,n}];
%t T[n_, k_, m_] = If[n==1, 2, Binomial[n, k]*(m+1)^n*t[n, m]*t[k, n-m]/n!];
%t Table[T[n, k, 1], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 09 2021 *)
%o (Sage)
%o def t(n, m): return (1/m^n)*product( (m+1)^j -1 for j in (1..n) )
%o def T(n,k,m): return 2 if n==1 else binomial(n,k)*(m+1)^n*t(n,m)*t(k,n-m)/factorial(n)
%o flatten([[T(n,k,1) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jul 09 2021
%K nonn,tabl,easy,less
%O 0,2
%A _Roger L. Bagula_, Feb 26 2009
%E Edited by _G. C. Greubel_, Jul 09 2021
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