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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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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EXAMPLE
| Rows begin:
[1 1 1 1 1 1 1 ...]
[1 4 10 20 35 56 84 ...]
[1 10 48 162 441 1036 2184 ...]
[1 20 162 841 3314 10786 30460 ...]
[1 35 441 3314 18004 77952 284880 ...]
[1 56 1036 10786 77952 435654 2007456 ...]
[1 84 2184 30460 284880 2007456 11427992 ...]
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PROG
| (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])); )); )); ))}
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CROSSREFS
| Cf. A089448 (diagonal), A089449 (antidiagonal sums), A086617, A088925, A002293.
Sequence in context: A109955 A174043 A175124 * A082680 A056939 A202924
Adjacent sequences: A089444 A089445 A089446 * A089448 A089449 A089450
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Nov 02 2003
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