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.

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

0,2

Links

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.

Formula

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

Example

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 +...

Mathematica

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 *)

Crossrefs

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

Sequence in context: A126566 A112808 A369158 * A110489 A005425 A035349

Adjacent sequences: A088924 A088925 A088926 * A088928 A088929 A088930

Keyword

nonn

Author

Paul D. Hanna, Oct 23 2003

Status

approved