A086621 Main diagonal of square table A086620 of 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)^2.

1, 3, 14, 79, 504, 3514, 26172, 204831, 1664696, 13930840, 119312544, 1041227642, 9228614836, 82867255956, 752405060536, 6897376441167, 63760133568096, 593763928313128, 5565678569009328, 52475976165495960, 497376657383374560, 4736680863568248480, 45304174896889357440

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Formula

a(n) = Sum_{j=0..n} binomial(n,j) * binomial(2*n-j,n) * binomial(2*j,j)/(j+1). - Andrew Howroyd, Apr 11 2021

D-finite with recurrence n*(n-1)*(n+1)^2*a(n) -2*n*(n-1) *(4*n+3) *(2*n-1) *a(n-1) +4*(n-1) *(16*n^3-20*n^2-13*n+14) *a(n-2) -4*(n-2) *(4*n-9) *(4*n-3) *(n+1) *a(n-3)=0. - R. J. Mathar, Nov 02 2021

a(n) ~ sqrt(5) * 2^(2*n - 3/2) * phi^(2*n + 5/2) / (Pi * n^2), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 19 2021

Maple

re:= sumtools:-sumrecursion(binomial(n, j) * binomial(2*n-j, n) * binomial(2*j, j)/(j+1), j, a(n)); # re = Mathar's recurrence
f:= gfun:-rectoproc({re = 0, a(0)=1, a(1)=3, a(2)=14}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Oct 23 2022

Prog

(PARI) a(n) = {sum(j=0, n, binomial(n, j)*binomial(2*n-j, n)*binomial(2*j, j)/(j+1))} \\ Andrew Howroyd, Apr 11 2021

Crossrefs

Cf. A086620 (table), A086622 (antidiagonal sums).

Sequence in context: A330074 A059276 A003169 * A020089 A218677 A353079

Adjacent sequences: A086618 A086619 A086620 * A086622 A086623 A086624

Keyword

nonn

Author

Paul D. Hanna, Jul 24 2003

Extensions

Terms a(18) and beyond from Andrew Howroyd, Apr 11 2021

Status

approved