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A331515 Expansion of 1/(1 - 8*x + 4*x^2)^(3/2).

%I A331515 #56 Aug 21 2025 00:09:41
%S A331515 1,12,114,1000,8430,69384,561988,4499856,35719830,281634760,
%T A331515 2208564732,17242680624,134118558028,1039939550160,8041848166920,
%U A331515 62042202765856,477670318108902,3670988584476744,28166853684793420,215807899372086000,1651323989374972836
%N A331515 Expansion of 1/(1 - 8*x + 4*x^2)^(3/2).
%H A331515 Seiichi Manyama, <a href="/A331515/b331515.txt">Table of n, a(n) for n = 0..1000</a>
%F A331515 a(n) = Sum_{k=1..n+1} 2^(n-k) * k * binomial(n+1,k) * binomial(n+1+k,k).
%F A331515 n * a(n) = 4 * (2*n+1) * a(n-1) - 4 * (n+1) * a(n-2) for n>1.
%F A331515 a(n) = ((n+2)/2) * Sum_{k=0..n} 3^k * binomial(n+1,k) * binomial(n+1,k+1).
%F A331515 a(n) ~ 2^(n - 1/2) * (2 + sqrt(3))^(n + 3/2) * sqrt(n) / (3^(3/4) * sqrt(Pi)). - _Vaclav Kotesovec_, Jan 26 2020
%F A331515 From _Seiichi Manyama_, Aug 20 2025: (Start)
%F A331515 a(n) = binomial(n+2,2) * A007564(n+1).
%F A331515 a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A331515 a(n) = Sum_{k=0..n} 2^k * (-1/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)
%t A331515 a[n_] := Sum[2^(n - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n + 1}]; Array[a, 21, 0] (* _Amiram Eldar_, Jan 20 2020 *)
%o A331515 (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(3/2))
%o A331515 (PARI) a(n) = sum(k=1, n+1, 2^(n-k)*k*binomial(n+1, k)*binomial(n+1+k, k));
%o A331515 (Magma) R<x>:=PowerSeriesRing(Rationals(), 21); Coefficients(R!( 1/(1 - 8*x + 4*x^2)^(3/2))); // _Marius A. Burtea_, Jan 20 2020
%o A331515 (Magma) [&+[2^(n-k)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..21]]; // _Marius A. Burtea_, Jan 20 2020
%Y A331515 Column 4 of A331514.
%Y A331515 Cf. A007564, A069835, A385728.
%K A331515 nonn
%O A331515 0,2
%A A331515 _Seiichi Manyama_, Jan 19 2020