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

0,5

Links

Table of n, a(n) for n=0..48.

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

Adjacent sequences: A140133 A140134 A140135 * A140137 A140138 A140139

Keyword

easy,nonn,tabf

Author

Paul Barry, May 09 2008

Status

approved