A157285 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.

1, 2, 2, 6, 12, 18, 28, 84, 336, 1456, 210, 840, 6300, 88200, 1874250, 2604, 13020, 156240, 4843440, 377788320, 59010535584, 54684, 328104, 5741820, 329197680, 63946649340, 39774815889480, 61856467844122980, 1984248, 13889736

Offset

0,2

Links

G. C. Greubel, Rows n = 0..30 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 preprint arXiv:0708.2349 [math.PR], 2007.

Formula

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!.

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

Example

Triangle begins as:
     1;
     2,     2;
     6,    12,     18;
    28,    84,    336,    1456;
   210,   840,   6300,   88200,   1874250;
  2604, 13020, 156240, 4843440, 377788320, 59010535584;

Mathematica

(* First program *)
t[n_, k_] = If[k==0, n!, Product[Sum[(k+1)^i, {i, 0, j-1}], {j, n}]];
T[n_, k_, m_] = If[n==0, 1, ((m+1)^n*t[n, m]*t[k, n-m])/(k!*(n-k)!)];
Flatten@Table[T[n, k, 1], {n, 0, 10}, {k, 0, n}]
(* Second program *)
t[n_, m_] = (1/m^n)*Product[(m+1)^j - 1, {j, n}];
T[n_, k_, m_] = If[n==1, 2, Binomial[n, k]*(m+1)^n*t[n, m]*t[k, n-m]/n!];
Table[T[n, k, 1], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 09 2021 *)

Prog

(Sage)
def t(n, m): return (1/m^n)*product( (m+1)^j -1 for j in (1..n) )
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)
flatten([[T(n, k, 1) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jul 09 2021

Crossrefs

Sequence in context: A290518 A242882 A361426 * A320068 A345988 A275439

Adjacent sequences: A157282 A157283 A157284 * A157286 A157287 A157288

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Jul 09 2021

Status

approved