A089448 -id:A089448 - cp's OEIS frontend

A089447 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies: f(x,y) = g(x,y) + xyf(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)g(x,y) + xyg(x,y)^2 = 0.

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

Offset: 0

Paul D. Hanna, Nov 02 2003

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.

			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, ...]; ...

Cf. A089448 (diagonal), A089449 (antidiagonal sums), A086617, A088925, A002293.

{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}

A089449 Antidiagonal sums of square table A089447, which lists the coefficients of x^ny^k in f(x,y) that satisfies: f(x,y) = g(x,y) + xyf(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)g(x,y) + xyg(x,y)^2 = 0.

Original entry on oeis.org

1, 2, 6, 22, 90, 396, 1837, 8870, 44186, 225628, 1175322, 6222788, 33392644, 181216728, 992829379, 5483790870, 30502513970, 170705626308, 960498281302, 5430200987260, 30830681187480, 175715526842056, 1004931956037782

Offset: 0

Paul D. Hanna, Nov 02 2003

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A089447 (table), A089448 (diagonal), A002293.

G.f.: A(x) = sum(n>=0, Catalan(n+1)*x^n) + x^2*A(x)^4, where Catalan(n)=(2n)!/(n!*(n+1)!).
From Vaclav Kotesovec, Oct 10 2020: (Start)
G.f.: A(x) = (1 - Sqrt[1-4*x] - 2*x)/(2*x^2) + x^2*A(x)^4.
a(n) ~ sqrt(11) * 3^(15/2 + 3*n) / ((8 + 3*sqrt(3) + 4*sqrt(4 + 3*sqrt(3))) * sqrt((2519 + 528*sqrt(3) + 2*sqrt(1484692 + 881529*sqrt(3))) * Pi) * n^(3/2) * 2^(2*n + 5/2) * (-32 - 18*sqrt(3) + sqrt(1996 + 1233*sqrt(3)))^(n+2)). (End)

cp's OEIS Frontend

A089447 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies: f(x,y) = g(x,y) + xyf(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)g(x,y) + xyg(x,y)^2 = 0.

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A089449 Antidiagonal sums of square table A089447, which lists the coefficients of x^ny^k in f(x,y) that satisfies: f(x,y) = g(x,y) + xyf(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)g(x,y) + xyg(x,y)^2 = 0.

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cp's OEIS Frontend

A089447 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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PARI

A089449 Antidiagonal sums of square table A089447, which lists the coefficients 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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A089447 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies: f(x,y) = g(x,y) + xyf(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)g(x,y) + xyg(x,y)^2 = 0.

A089449 Antidiagonal sums of square table A089447, which lists the coefficients of x^ny^k in f(x,y) that satisfies: f(x,y) = g(x,y) + xyf(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)g(x,y) + xyg(x,y)^2 = 0.