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EXAMPLE
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Square array begins:
(1),(1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1,(2),(3), 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...;
1, 5,(10),(16), 23, 31, 40, 50, 61, 73, 86, 100, 115, 131, 148, ...;
1, 6, 29,(60),(100), 150, 211, 284, 370, 470, 585, 716, 864, ...;
1, 7, 36, 186,(397),(681), 1051, 1521, 2106, 2822, 3686, 4716, ...;
1, 8, 44, 230, 1281,(2802),(4908), 7730, 11416, 16132, 22063, ...;
1, 9, 53, 283, 1564, 9294,(20710),(36842), 58905, 88319, 126730, ...;
1, 10, 63, 346, 1910, 11204, 70109,(158428),(285158), 461190, ...;
1, 11, 74, 420, 2330, 13534, 83643, 544833,(1244413),(2260257), ...;
...
For each row, remove the terms along the diagonals (in parenthesis),
and then take partial sums to obtain the next row.
GENERATING FUNCTIONS.
The g.f. of n-th lower diagonal equals D(x)*F(x)^2*C(x)^n and
the g.f. of n-th upper diagonal equals D(x)*F(x)^n,
where D(x) is g.f. of main diagonal (A137571):
[1, 2, 10, 60, 397, 2802, 20710, 158428, 1244413, 9980220, ...]
defined by:
D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].
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PROGRAM
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(PARI) {T(n, k)=if(k<0, 0, if(n==0, 1, T(n, k-1) + if(n-1>k, T(n-1, k), T(n-1, k+2))))} (PARI) /* Using Formula for G.F.: */ {T(n, k)=local(m=max(n, k)+1, C, F, D, A); C=subst(Ser(vector(m, r, binomial(2*r-2, r-1)/r)), x, x*y); F=subst(Ser(vector(m, r, binomial(4*r-4, r-1)/(3*r-2))), x, x*y); D=1/(1-x*y*C*F^2-x*y*F^3); A=D*(1/(1-y*F) + x*C*F^2/(1-x*C)); polcoeff(polcoeff(A+O(x^m), n, x)+O(y^m), k, y)}
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