A062985 Generalized Catalan array FS(5; n,r).

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 5, 5, 1, 3, 6, 10, 15, 20, 25, 30, 35, 35, 35, 35, 35, 1, 4, 10, 20, 35, 55, 80, 110, 145, 180, 215, 250, 285, 285, 285, 285, 285, 1, 5, 15, 35, 70, 125, 205, 315, 460, 640, 855, 1105

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=5.

The step width sequence of this staircase array is [1,4,4,4,....], i.e. the degree of the row polynomials is [0,4,8,12,...]= A008586.

The columns r=0..7 (without leading zeros) give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A000292 (tetrahedral), A000332(4+n), A062988-A062990.

Examples

			{1}; {1,1,1,1,1}; {1,2,3,4,5,5,5,5,5}; ...; N(5; 1,x)=5-10*x+10*x^2-5*x^3+x^4.
		

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>4*n; a(n, r)=a(n, r-1)+a(n-1, r) else.

G.f. for column r=4*k+j, k >= 0, j=1, 2, 3, 4: (x^(k+1))*N(5; k, x)/(1-x)^(4*k+1+j), with row polynomials of array A062986.