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A370616 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2) )^n.

%I A370616 #18 Jul 30 2025 05:38:49
%S A370616 1,0,2,3,14,35,125,371,1238,3909,12847,41580,136577,447187,1473341,
%T A370616 4855703,16053830,53138243,176233967,585202261,1945964079,6478043120,
%U A370616 21588979876,72016891508,240452892569,803489258285,2686964354375,8991840800136,30110638705889
%N A370616 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2) )^n.
%F A370616 a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(n-k-1,n-2*k).
%F A370616 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x-x^2) / (1-x) ).
%F A370616 From _Peter Bala_, 26 Jul 2025: (Start)
%F A370616 a(n) = n * hypergeom([1 + n, 1 - n/2, 3/2 - n/2], [2, 2 - n], -4) for n >= 3.
%F A370616 P-recursive: 5*n*(74*n^3-493*n^2+1075*n-766)*(n-1)*a(n) = 2*(n-1)*(296*n^4-2120*n^3+5393*n^2-5716*n+2100)*a(n-1) + 2*(1184*n^5-10256*n^4+34088*n^3-53995*n^2+40397*n-11250)*a(n-2) - 2*(n-3)*(2*n-5)*(74*n^3-271*n^2+311*n-110)*a(n-3) with a(0) = 1, a(1) = 0 and a(2) = 2.
%F A370616 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k. (End)
%F A370616 a(n) ~ sqrt(1/12 + sqrt(10/37)*(sin(arcsin((13*sqrt(37/10))/40)/3)/3)) * (8*((1 + sqrt(34)*cos(arccos(2461/(1088*sqrt(34)))/3))/15))^n / sqrt(Pi*n). - _Vaclav Kotesovec_, Jul 30 2025
%t A370616 Table[Sum[Binomial[-1 - k + n, -2*k + n] Binomial[-1 + k + n, k], {k, 0, n/2}], {n, 0, 30}] (* _Vaclav Kotesovec_, Jul 30 2025 *)
%o A370616 (PARI) a(n, s=2, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));
%Y A370616 Cf. A046736, A213684, A104507, A246437, A386548.
%K A370616 nonn
%O A370616 0,3
%A A370616 _Seiichi Manyama_, Apr 30 2024