A209200 - OEIS

%I #15 Jul 18 2013 11:29:59

%S 1,4,20,112,680,4384,29536,205440,1462368,10587520,77633920,574845440,

%T 4289409280,32206976000,243074083840,1842511532032,14018197145088,

%U 106996519311360,818973463721984,6284217844736000,48327723087278080,372397083591557120

%N G.f.: (1-4*x)^(-1/2) * (1-8*x)^(-1/4).

%C Equals the convolution of sequences A000984 and A004981.

%C The sequences A000984 and A004981 are related by the identity:

%C Sum_{n>=0} A000984(n)^3 *x^n = ( Sum_{n>=0} A004981(n)^2 *x^n )^2.

%H Vincenzo Librandi, <a href="/A209200/b209200.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{k=0..n} A000984(n-k)*A004981(k).

%F Recurrence: n*a(n) = 4*(3*n-2)*a(n-1) - 8*(4*n-5)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012

%F a(n) ~ Gamma(3/4)*8^n/(Pi*n^(3/4)). - _Vaclav Kotesovec_, Oct 20 2012

%e G.f.: A(x) = 1 + 4*x + 60*x^2 + 1200*x^3 + 27300*x^4 + 668304*x^5 +...

%e This sequence equals the convolution of the sequences:

%e A000984 = [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...], and

%e A004981 = [1, 2, 10, 60, 390, 2652, 18564, 132600, 961350, ...].

%e Related sequences:

%e A^2: [1, 8, 56, 384, 2656, 18688, 133888, 974848, 7194112, ...],

%e A^4: [1, 16, 176, 1664, 14592, 122880, 1011712, 8224768, ...].

%t CoefficientList[Series[(1-4*x)^(-1/2)*(1-8*x)^(-1/4), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)

%o (PARI) {a(n)=polcoeff((1-4*x +x*O(x^n))^(-1/2)*(1-8*x +x*O(x^n))^(-1/4),n)}

%o (PARI) {A000984(n)=polcoeff((1-4*x +x*O(x^n))^(-1/2),n)}

%o {A004981(n)=polcoeff((1-8*x +x*O(x^n))^(-1/4),n)}

%o {a(n)=sum(k=0,n,A000984(n-k)*A004981(k))}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A000984, A004981.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 06 2012