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%I #11 Oct 30 2018 20:18:01
%S 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,
%T 74,96,118,140,140,140,140,1,5,15,35,69,121,195,291,409,549,689,829,
%U 969,969,969,969,1,6,21,56,125
%N Generalized Catalan array FS(4; n,r).
%C In the Frey-Sellers reference this array is called {n over r}_{m-1}, with m=4.
%C 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.
%C The columns r=0..6 give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A000292 (tetrahedral), A063258, A027659, A062966.
%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>3*n; a(n, r)=a(n, r-1)+a(n-1, r) else.
%F 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.
%e {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).
%K nonn,easy,tabf
%O 0,7
%A _Wolfdieter Lang_, Jul 12 2001
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