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A385728 Expansion of 1/((1-2*x) * (1-6*x))^(3/2).

%I A385728 #49 Aug 25 2025 04:37:47
%S A385728 1,12,102,760,5310,35784,235788,1530288,9824310,62557000,395797908,
%T A385728 2491381776,15616141996,97537784400,607391245080,3772617319008,
%U A385728 23379854507046,144605546475336,892834113930180,5504041611527760,33883431379007364,208327771987901808
%N A385728 Expansion of 1/((1-2*x) * (1-6*x))^(3/2).
%H A385728 Paolo Xausa, <a href="/A385728/b385728.txt">Table of n, a(n) for n = 0..1000</a>
%F A385728 n*a(n) = (8*n+4)*a(n-1) - 12*(n+1)*a(n-2) for n > 1.
%F A385728 a(n) = (1/2)^n * Sum_{k=0..n} 3^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A385728 a(n) = Sum_{k=0..n} 2^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A385728 a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A385728 a(n) = binomial(n+2,2) * A005572(n).
%F A385728 a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A385728 a(n) = Sum_{k=0..n} 2^k * (-3/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
%F A385728 a(n) ~ sqrt(n) * 2^(n - 1/2) * 3^(n + 3/2) / sqrt(Pi). - _Vaclav Kotesovec_, Aug 21 2025
%t A385728 Module[{a, n}, RecurrenceTable[{a[n] == ((8*n+4)*a[n-1] - 12*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 12}, a, {n, 0, 25}]] (* _Paolo Xausa_, Aug 21 2025 *)
%t A385728 CoefficientList[Series[ 1/((1-2*x)*(1-6*x))^(3/2),{x,0,33}],x] (* _Vincenzo Librandi_, Aug 22 2025 *)
%o A385728 (PARI) my(N=30, x='x+O('x^N)); Vec(1/((1-2*x)*(1-6*x))^(3/2))
%o A385728 (Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1 / ((1 - 2*x) * (1 - 6*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // _Vincenzo Librandi_, Aug 22 2025
%Y A385728 Cf. A385563, A385813.
%Y A385728 Cf. A005572, A081671, A331515.
%K A385728 nonn,changed
%O A385728 0,2
%A A385728 _Seiichi Manyama_, Aug 19 2025