A088925 -id:A088925 - cp's OEIS frontend

A088927 Antidiagonal sums of table A088925, which lists coefficients T(n,k) of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^3.

1, 2, 5, 14, 43, 142, 496, 1808, 6807, 26270, 103357, 412942, 1670572, 6828824, 28159880, 116997296, 489271039, 2057800158, 8698624303, 36936288650, 157474552403, 673830974654, 2892864930292, 12457038200008, 53789813903620

Offset: 0

Paul D. Hanna, Oct 23 2003

nonn

			A(x) = 1/(1-2x) + x^2*A(x)^3 since 1/(1-2x) = 1 + 2x + 4x^2 + 8x^3 +... and x^2*A(x)^3 = 1x^2 + 6x^3 + 27x^4 + 110x^5 +...

Michael De Vlieger, Table of n, a(n) for n = 0..300
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Cf. A088925 (table), A088926 (diagonal), A001764.

Table[Sum[Sum[Binomial[n, 2*i] * Binomial[n - 2*i, k - i] * (3*i)! / (i! * (2*i + 1)!), {i, 0, k}], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 10 2020 *)

a(n) = sum(k=0, n, sum(i=0, k, C(n, 2i)*C(n-2i, k-i)*A001764(i) )), where A001764(i)=(3i)!/[i!(2i+1)! ] (from Michael Somos).
G.f. satisfies A(x) = 1/(1-2x) + x^2*A(x)^3.
a(n) ~ (2 + 3*sqrt(3)/2)^(n + 3/2) / (3^(7/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 10 2020

A088926 Main diagonal of table A088925, which lists coefficients T(n,k) of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^3.

Original entry on oeis.org

1, 3, 21, 212, 2617, 36345, 544080, 8577378, 140456625, 2368062095, 40859183247, 718386164556, 12829418522056, 232153200359592, 4248457201595622, 78508329463480160, 1463164022514939392, 27474112707608092672

Offset: 0

Paul D. Hanna, Oct 23 2003

nonn

The g.f. for A001764 satisfies: g(x) = 1 + x*g(x)^3.

Cf. A088925 (table), A088927 (antidiagonal sums), A001764.

Table[Sum[Binomial[2*n, 2*i] * Binomial[2*n - 2*i, n - i]*(3*i)!/(i!*(2*i + 1)!), {i, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 10 2020 *)

a(n) = sum(i=0, n, C(2n, 2i)*C(2n-2i, n-i)*A001764(i) ), where A001764(i)=(3i)!/[i!(2i+1)! ] (from Michael Somos).

a(n) ~ (4 + 3*sqrt(3))^(2*n + 2) / (Pi * 3^(7/4) * n^2 * 2^(2*n + 4)). - Vaclav Kotesovec, Oct 10 2020

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.

Original entry on oeis.org

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}

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A088927 Antidiagonal sums of table A088925, which lists coefficients T(n,k) of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^3.

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A088926 Main diagonal of table A088925, which lists coefficients T(n,k) of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^3.

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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.

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cp's OEIS Frontend

A088927 Antidiagonal sums of table A088925, which lists coefficients T(n,k) of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^3.

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Mathematica

Formula

A088926 Main diagonal of table A088925, which lists coefficients T(n,k) of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^3.

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Mathematica

Formula

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.

Views

Author

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Comments

Examples

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Programs

PARI

A088927 Antidiagonal sums of table A088925, which lists coefficients T(n,k) of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^3.

A088926 Main diagonal of table A088925, which lists coefficients T(n,k) of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^3.

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.