login
A140136
Numerator coefficients for generators of lattice path enumeration square array A111910.
2
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
0,5
LINKS
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.
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).
EXAMPLE
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;
...
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 *)
CROSSREFS
Row sums are A006335.
Cf. A111910.
Sequence in context: A242914 A046542 A172351 * A281123 A171707 A156722
KEYWORD
easy,nonn,tabf
AUTHOR
Paul Barry, May 09 2008
STATUS
approved