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A386897 a(n) = 4^n * binomial(5*n/2,n).

%I A386897 #37 Aug 26 2025 10:05:10
%S A386897 1,10,160,2860,53760,1040060,20500480,409404600,8255569920,
%T A386897 167718033340,3427543285760,70384350760360,1451115518361600,
%U A386897 30018413447053080,622759359440486400,12951795276279787760,269947721071617638400,5637113741080428839100
%N A386897 a(n) = 4^n * binomial(5*n/2,n).
%H A386897 Paolo Xausa, <a href="/A386897/b386897.txt">Table of n, a(n) for n = 0..700</a>
%F A386897 a(n) == 0 (mod 10) for n > 0.
%F A386897 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(4*n-k,n-k).
%F A386897 a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(3*n+1).
%F A386897 a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(3*n+1)).
%F A386897 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(2*n-k,n-k).
%F A386897 a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k,k) * binomial(2*n-k,n-k).
%F A386897 a(n) = [x^n] 1/(1-4*x)^(3*n/2+1).
%F A386897 a(n) = [x^n] (1+4*x)^(5*n/2).
%F A386897 a(n) ~ 2^(n - 1/2) * 5^((5*n+1)/2) / (sqrt(Pi*n) * 3^((3*n+1)/2)). - _Vaclav Kotesovec_, Aug 07 2025
%F A386897 D-finite with recurrence 3*n*(n-1)*(3*n-4) *(3*n-2)*a(n) -20*(5*n-4) *(5*n-8)*(5*n-2) *(5*n-6)*a(n-2)=0. - _R. J. Mathar_, Aug 21 2025
%F A386897 O.g.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/3, 1/2, 2/3], (12500*x^2)/27) + 10*x*hypergeom([7/10, 9/10, 11/10, 13/10], [5/6, 7/6, 3/2], (12500*x^2)/27). - _Karol A. Penson_, Aug 26 2025
%t A386897 Table[Sum[2^k *(-1)^(n-k)*Binomial[5*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 07 2025 *)
%t A386897 A386897[n_] := 4^n*Binomial[5*n/2, n]; Array[A386897, 20, 0] (* _Paolo Xausa_, Aug 26 2025 *)
%o A386897 (PARI) a(n) = 4^n*binomial(5*n/2, n);
%Y A386897 Cf. A386812, A386895, A386896, A386898.
%Y A386897 Cf. A000302, A098430, A244038.
%K A386897 nonn,changed
%O A386897 0,2
%A A386897 _Seiichi Manyama_, Aug 07 2025