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%I #28 Sep 04 2019 08:13:15
%S 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 6720,15288,15288,6720,1212,75,1,1,121,3223,30723,127908,255816,
%U 255816,127908,30723,3223,121,1,1,182,7371,109538,737737,2510508
%N Numerator coefficients for generators of lattice path enumeration square array A111910.
%H 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 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.
%F (Sum_{k=0..n} T(n,k) * x^k) / (1-x)^(3*n+1) generates row n of A111910.
%F 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 Triangle begins:
%e 1;
%e 1, 1;
%e 1, 7, 7, 1;
%e 1, 20, 75, 75, 20, 1;
%e 1, 42, 364, 1001, 1001, 364, 42, 1;
%e 1, 75, 1212, 6720, 15288, 15288, 6720, 1212, 75, 1;
%e ...
%t 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 Join[{{1}}, Table[T[n, k], {k, 2, 8}, {n, 1, 2 k - 2}]] // Flatten (* _Peter Luschny_, Sep 04 2019 *)
%Y Row sums are A006335.
%Y Cf. A111910.
%K easy,nonn,tabf
%O 0,5
%A _Paul Barry_, May 09 2008
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