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

%I A140136 #36 Jan 17 2025 09:27:54
%S A140136 1,1,1,1,7,7,1,1,20,75,75,20,1,1,42,364,1001,1001,364,42,1,1,75,1212,
%T A140136 6720,15288,15288,6720,1212,75,1,1,121,3223,30723,127908,255816,
%U A140136 255816,127908,30723,3223,121,1,1,182,7371,109538,737737,2510508
%N A140136 Numerator coefficients for generators of lattice path enumeration square array A111910.
%H A140136 Michael De Vlieger, <a href="/A140136/b140136.txt">Table of n, a(n) for n = 0..10100</a> (rows 0..100, flattened)
%H A140136 G. Kreweras, <a href="http://www.numdam.org/article/BURO_1965__6__9_0.pdf">Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers</a>, Cahiers du B.U.R.O. 6 (1965), 9-107; see p. 93.
%H A140136 G. Kreweras and H. Niederhausen, <a href="http://dx.doi.org/10.1016/S0195-6698(81)80020-0">Solution of an enumerative problem connected with lattice paths</a>, European J. Combin. 2 (1981), 55-60; see p. 60.
%H A140136 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 8.
%F A140136 (Sum_{k=0..n} T(n,k) * x^k) / (1-x)^(3*n+1) generates row n of A111910.
%F A140136 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).
%e A140136 Triangle begins:
%e A140136   1;
%e A140136   1,  1;
%e A140136   1,  7,    7,     1;
%e A140136   1, 20,   75,    75,    20,     1;
%e A140136   1, 42,  364,  1001,  1001,   364,   42,    1;
%e A140136   1, 75, 1212,  6720, 15288, 15288, 6720, 1212, 75, 1;
%e A140136   ...
%t A140136 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)!);
%t A140136 Join[{{1}}, Table[T[n, k], {k, 2, 8}, {n, 1, 2 k - 2}]] // Flatten (* _Peter Luschny_, Sep 04 2019 *)
%Y A140136 Row sums are A006335.
%Y A140136 Cf. A111910.
%K A140136 easy,nonn,tabf
%O A140136 0,5
%A A140136 _Paul Barry_, May 09 2008