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A062750 Generalized Catalan array FS(4; n,r). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
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.
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.
EXAMPLE
{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).
CROSSREFS
Sequence in context: A171502 A005102 A030241 * A368420 A193669 A065686
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Jul 12 2001
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)