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A288470 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n,2*k).

%I A288470 #26 Aug 31 2025 09:10:06
%S A288470 1,2,14,92,646,4652,34124,253528,1901638,14368844,109208164,833981128,
%T A288470 6394017436,49185717752,379438594136,2934361958192,22741538394694,
%U A288470 176582855512588,1373431963785332,10698376362421096,83447762846703796,651690159076273192,5095051571420324264,39874449115469939152,312350761370734541596
%N A288470 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n,2*k).
%C A288470 Row sums of A155495.
%C A288470 a(n) is the constant term in the expansion of (-1 + (1 + x + 1/x)^2)^n. - _Seiichi Manyama_, Nov 21 2019
%H A288470 Seiichi Manyama, <a href="/A288470/b288470.txt">Table of n, a(n) for n = 0..1109</a>
%F A288470 a(n) = hypergeom([-n,-n,1/2-n],[1/2,1],-1).
%F A288470 n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1).
%F A288470 G.f.: sqrt((1-2*x+sqrt(1-8*x))/(2*(1-7*x-8*x^2))).
%F A288470 a(n) ~ 8^n / sqrt(3*Pi*n). - _Vaclav Kotesovec_, Nov 27 2017
%F A288470 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(3*n-2*k-1,n-2*k). - _Seiichi Manyama_, Feb 13 2024
%F A288470 From _Peter Bala_, Aug 30 2025: (Start)
%F A288470 n*(2*n - 1)*(3*n - 4)*a(n) = 2*(21*n^3 - 49*n^2 + 33*n - 6)*a(n-1) + 8*(n - 1)*(3*n - 1)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2.
%F A288470 a(n) = Sum_{0 <= i, j <= n/2} binomial(2*n, j)*binomial(2*n+i-1, i)*binomial(2*n, n- 2*i-2*j) (verified to satisfy the above second-order recurrence using the MulZeil procedure in Doron Zeilberger's MultiZeilberger Maple package).
%F A288470 Equivalently, a(n) = [x^n] ( (1 + x + x^2 + x^3)/(1 - x^2) )^(2*n). Cf. A240688.
%F A288470 The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k)  for all primes p and positive integers n and k.
%F A288470 Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)
%p A288470 f:= gfun:-rectoproc({n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1), a(0)=1,a(1)=2, a(2)=14},a(n),remember):
%p A288470 map(f, [$0..30]);
%t A288470 Table[Sum[Binomial[n, k] Binomial[2 n, 2 k], {k, 0, n}], {n, 0, 24}] (* _Michael De Vlieger_, Jun 09 2017 *)
%o A288470 (PARI) {a(n) = polcoef((-1+(1+x+1/x)^2)^n, 0)} \\ _Seiichi Manyama_, Nov 21 2019
%Y A288470 Cf. A094061, A155495, A218045.
%K A288470 nonn,easy,changed
%O A288470 0,2
%A A288470 _Robert Israel_, Jun 09 2017