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A062985 Generalized Catalan array FS(5; n,r). 6

%I #10 Oct 30 2018 12:52:41

%S 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,

%T 4,10,20,35,55,80,110,145,180,215,250,285,285,285,285,285,1,5,15,35,

%U 70,125,205,315,460,640,855,1105

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

%C In the Frey-Sellers reference this array is called {n over r}_{m-1}, with m=5.

%C 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.

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

%H D. D. Frey and J. A. Sellers, <a href="http://www.fq.math.ca/Scanned/39-2/frey.pdf">Generalizing Bailey's generalization of the Catalan numbers</a>, The Fibonacci Quarterly, 39 (2001) 142-148.

%F 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.

%F 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.

%e {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.

%K nonn,easy,tabf

%O 0,8

%A _Wolfdieter Lang_, Jul 12 2001

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