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A089447
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Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies: f(x,y) = g(x,y) + xy*f(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)*g(x,y) + xy*g(x,y)^2 = 0.
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3
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1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 48, 20, 1, 1, 35, 162, 162, 35, 1, 1, 56, 441, 841, 441, 56, 1, 1, 84, 1036, 3314, 3314, 1036, 84, 1, 1, 120, 2184, 10786, 18004, 10786, 2184, 120, 1, 1, 165, 4236, 30460, 77952, 77952, 30460, 4236, 165, 1, 1, 220, 7689, 77044
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OFFSET
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0,5
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COMMENTS
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Explicitly, g(x,y) = ((1-x-y)+sqrt((1-x-y)^2-4xy))/(2xy) = sum(n>=0, sum(k>=0, N(n,k)*x^n*y^k), where N(n,k) are the Narayana numbers: N(n,k) = C(n+k,k)*C(n+k+2,k+1)/(n+k+2). This array is directly related to sequence A002293, which has a g.f. h(x) that satisfies h(x) = 1 + x*h(x)^4. The inverse binomial transform of the rows grows by three terms per row.
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LINKS
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EXAMPLE
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Rows begin:
[1, 1, 1, 1, 1, 1, 1, 1, ...];
[1, 4, 10, 20, 35, 56, 84, 120, ...];
[1, 10, 48, 162, 441, 1036, 2184, 4236, ...];
[1, 20, 162, 841, 3314, 10786, 30460, 77044, ...];
[1, 35, 441, 3314, 18004, 77952, 284880, 912042, ...];
[1, 56, 1036, 10786, 77952, 435654, 2007456, 7951674, ...];
[1, 84, 2184, 30460, 284880, 2007456, 11427992, 55009548, ...];
[1, 120, 4236, 77044, 912042, 7951674, 55009548, 317112363, ...];
[1, 165, 7689, 178387, 2624453, 27870393, 231114465, 1576219474, ...]; ...
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PROG
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(PARI) {L=10; T=matrix(L, L, n, k, 1); for(n=1, L-1, for(k=1, L-1, T[n+1, k+1]=binomial(n+k, k)*binomial(n+k+2, k+1)/(n+k+2)+ sum(j3=1, k, sum(i3=1, n, T[n-i3+1, k-j3+1]* sum(j2=1, j3, sum(i2=1, i3, T[i3-i2+1, j3-j2+1]* sum(j1=1, j2, sum(i1=1, i2, T[i2-i1+1, j2-j1+1]*T[i1, j1])); )); )); )); T}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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