A086621 - OEIS

%I #13 Oct 23 2022 23:29:34

%S 1,3,14,79,504,3514,26172,204831,1664696,13930840,119312544,

%T 1041227642,9228614836,82867255956,752405060536,6897376441167,

%U 63760133568096,593763928313128,5565678569009328,52475976165495960,497376657383374560,4736680863568248480,45304174896889357440

%N 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.

%H Andrew Howroyd, <a href="/A086621/b086621.txt">Table of n, a(n) for n = 0..500</a>

%F 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

%F 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

%F 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

%p re:= sumtools:-sumrecursion(binomial(n,j) * binomial(2*n-j,n) * binomial(2*j,j)/(j+1),j,a(n)); # re = Mathar's recurrence

%p 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

%o (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

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 24 2003

%E Terms a(18) and beyond from _Andrew Howroyd_, Apr 11 2021