OFFSET
0,5
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.
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).
From Sergii Voloshyn, May 24 2026: (Start)
G.f.: F(x,y) = y^2*x^2/(1 - x^2) + y^4*x^2*(1 + x^2)/(1 - x^2)^4 + y^6*x^2*(1 + 7*x^2 + 7*x^4 + x^6)/(1 - x^2)^7 is the solution of the differential equation x * d^3(F(x,y))/dx^3 = y * d^3(F(x,y))/dy^3.
Let E be the operator x*D*D*D, where D denotes the derivative operator d/dx. Then (1/((2*n + 2)*(2*n + 1)*(2*n))) * E^n(x^2/(1 - x^2)) = (row n generating polynomial)/(1 - x)^(3*n+1). For example, when n = 2 we have (1/24/120)*E^2(x^2/(1 - x^2)) = x^2*(1 + 7*x^2 + 7*x^4 + x^6)/(1 - x^2)^7. (End)
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
KEYWORD
easy,nonn,tabf,changed
AUTHOR
Paul Barry, May 09 2008
STATUS
approved