A062750 Generalized Catalan array FS(4; n,r).

1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 1, 3, 6, 10, 14, 18, 22, 22, 22, 22, 1, 4, 10, 20, 34, 52, 74, 96, 118, 140, 140, 140, 140, 1, 5, 15, 35, 69, 121, 195, 291, 409, 549, 689, 829, 969, 969, 969, 969, 1, 6, 21, 56, 125

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• nonn
• easy
• tabf

Comments

In the Frey-Sellers reference this array is called {n over r}_{m-1}, with m=4.

The step width sequence of this staircase array is [1,3,3,3,....], i.e. the degree of the row polynomials is [0,3,6,9,...]= A008585.

The columns r=0..6 give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A000292 (tetrahedral), A063258, A027659, A062966.

Examples

			{1}; {1,1,1,1}; {1,2,3,4,4,4,4}; {1,3,6,10,14,18,22,22,22,22}; ...; N(4; 1,x)=(2-x)*(2-2*x+x^2).
		

Links

• D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.

Formula

a(0, 0)=1, a(n, -1)=0, n >= 1; a(n, r)=0 if r>3*n; a(n, r)=a(n, r-1)+a(n-1, r) else.

G.f. for column r=3*k+j, k >= 0, j=1, 2, 3: (x^(k+1))*N(4; k, x)/(1-x)^(3*k+1+j), with the row polynomials N(4; k, x) of array A062751.