A188648 - OEIS

%I #46 Sep 06 2022 04:25:16

%S 1,2,11,63,376,2317,14545,92512,594169,3844787,25027296,163701327,

%T 1075049011,7083830648,46812088751,310118453573,2058919125662,

%U 13695571200353,91254952276859,608960974528058,4069232436916151

%N Binomial sums a(n) = Sum_{k=0..n} (binomial(2n-k,k))^2.

%C Central coefficients of A172991.

%C Bisection of A051286 (Whitney number of level n of the lattice of the ideals of the fence of order 2n). - _Paul D. Hanna_, Apr 07 2011

%H Seiichi Manyama, <a href="/A188648/b188648.txt">Table of n, a(n) for n = 0..1198</a> (terms 0..121 from Vincenzo Librandi)

%F G.f.: 1/2*(1/sqrt(1-2*sqrt(x)-x-2*x*sqrt(x)+x^2) + 1/sqrt(1+2*sqrt(x)-x+2*x*sqrt(x)+x^2)).

%F Recurrence: (n-2)*n*(2*n - 1)*(48*n^2 - 192*n + 169)*a(n) = (576*n^5 - 4032*n^4 + 10212*n^3 - 11414*n^2 + 5457*n - 849)*a(n-1) + 5*(2*n - 3)*(48*n^4 - 288*n^3 + 565*n^2 - 399*n + 64)*a(n-2) + (576*n^5 - 4608*n^4 + 13668*n^3 - 18286*n^2 + 10521*n - 1896)*a(n-3) - (n-3)*(n-1)*(2*n - 5)*(48*n^2 - 96*n + 25)*a(n-4). - _Vaclav Kotesovec_, Mar 02 2014

%F a(n) ~ phi^(4*n + 2) / (2^(3/2) * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Mar 02 2014, simplified Jan 13 2019

%F Conjecture: a(n) = hypergeom([-n,-n,n+1,n+1], [1/2,1/2,1], 1/16). - _Velin Yanev_, Oct 31 2019

%F a(n) = A051286(2*n). - _Mark van Hoeij_, Sep 05 2022

%t Table[Sum[Binomial[2n-k,k]^2,{k,0,n}],{n,0,20}]

%t Table[DifferenceRoot[Function[{y, m}, {4 (-m + n)^2 (-1 - 2 m + 2 n)^2 y[m] + (-5 m^2 - 18 m^3 - 17 m^4 + 12 m n + 56 m^2 n + 68 m^3 n - 8 n^2 - 56 m n^2 - 100 m^2 n^2 + 16 n^3 + 64 m n^3 - 16 n^4) y[1 + m] + (1 + m)^2 (-m + 2 n)^2 y[2 + m] == 0, y[0] == 0, y[1] == 1}]][n + 1], {n, 0, 20}] (* _Benedict W. J. Irwin_, Nov 03 2016 *)

%o (Maxima) makelist(sum(binomial(2*n-k,k)^2,k,0,n),n,0,20);

%o (PARI) {a(n) = sum(k=0, n, binomial(2*n-k, k)^2)} \\ _Seiichi Manyama_, Jan 13 2019

%Y Sum_{k=0..n} (binomial(2n-k,k))^b: A122367(n) = A001519(n+1) (b=1), this sequence (b=2).

%Y Cf. A054142, A172991, A051286.

%K nonn,easy

%O 0,2

%A _Emanuele Munarini_, Apr 07 2011