A140136 Numerator coefficients for generators of lattice path enumeration square array A111910.

1, 1, 1, 1, 7, 7, 1, 1, 20, 75, 75, 20, 1, 1, 42, 364, 1001, 1001, 364, 42, 1, 1, 75, 1212, 6720, 15288, 15288, 6720, 1212, 75, 1, 1, 121, 3223, 30723, 127908, 255816, 255816, 127908, 30723, 3223, 121, 1, 1, 182, 7371, 109538, 737737, 2510508

Offset: 0

Author

Paul Barry, May 09 2008

Keywords

• easy
• nonn
• tabf

Examples

			Triangle begins:
  1;
  1,  1;
  1,  7,    7,     1;
  1, 20,   75,    75,    20,     1;
  1, 42,  364,  1001,  1001,   364,   42,    1;
  1, 75, 1212,  6720, 15288, 15288, 6720, 1212, 75, 1;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10100 (rows 0..100, flattened)
• G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du B.U.R.O. 6 (1965), 9-107; see p. 93.
• G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin. 2 (1981), 55-60; see p. 60.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 8.

Crossrefs

Row sums are A006335.

Cf. A111910.

Programs

Mathematica

T[n_, k_] := ((k + n - 1)! (2 (k + n) - 3)! HypergeometricPFQ[{2 - 3 k, 1/2 - n, 1 - n, -n}, {1 - k - n, 3/2 - k - n, 2 - k - n}, 1])/(k! (2 k - 1)! n! (2 n - 1)!);
Join[{{1}}, Table[T[n, k], {k, 2, 8}, {n, 1, 2 k - 2}]] // Flatten (* Peter Luschny, Sep 04 2019 *)

Formula

(Sum_{k=0..n} T(n,k) * x^k) / (1-x)^(3*n+1) generates row n of A111910.

Triangle T(q,n), where T(n,q) = Sum_{j = 0..n} (-1)^j*C(3*q+1,j)*K(n-j,q) with K(p,q) = A111910(p,q).