%I #40 Mar 25 2022 09:13:35

%S 1,18,198,1716,12870,87516,554268,3325608,19122246,106234700,

%T 573667380,3024791640,15628090140,79342611480,396713057400,

%U 1957117749840,9540949030470,46021048264620,219878341708740,1041528987041400,4895186239094580,22844202449108040

%N Expansion of 1/(1-4*x)^(9/2).

%C Also convolution of A000984 with A038846, also convolution of A000302 with A020918, also convolution of A002457 with A038845, also convolution of A002697 with A002802. - _Rui Duarte_, Oct 08 2011

%H Vincenzo Librandi, <a href="/A020920/b020920.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = binomial(n+4, 4)*A000984(n+4)/A000984(4), where A000984 are the central binomial coefficients. - _Wolfdieter Lang_

%F a(n) = Sum_{ a+b+c+d+e+f+g+h+i=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g) *f(h)*f(i) with f(n)=A000984(n). - _Philippe Deléham_, Jan 22 2004

%F a(n) = A000332(n+4)*A000984(n+4)/70. - _Zerinvary Lajos_, May 05 2007

%F From _Rui Duarte_, Oct 08 2011: (Start)

%F a(n) = ((2n+7)(2n+5)(2n+3)(2n+1)/(7*5*3*1)) * binomial(2n, n).

%F a(n) = binomial(2n+8, 8) * binomial(2n, n) / binomial(n+4, 4).

%F a(n) = binomial(n+4, 4) * binomial(2n+8, n+4) / binomial(8, 4). (End)

%F Boas-Buck recurrence: a(n) = (18/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+4, 4). See a comment there.

%F From _Amiram Eldar_, Mar 25 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 1148/5 - 42*sqrt(3)*Pi.

%F Sum_{n>=0} (-1)^n/a(n) = 700*sqrt(5)*log(phi) - 11284/15, where phi is the golden ratio (A001622). (End)

%p seq(binomial(2*n+8, n+4)*binomial(n+4, n)/70, n=0..30); # _Zerinvary Lajos_, May 05 2007

%t CoefficientList[Series[1/(1-4x)^(9/2), {x,0,30}], x] (* _Vincenzo Librandi_, Jul 05 2013 *)

%o (Magma) [(2*n+7)*(2*n+5)*(2*n+3)*(2*n+1)*Binomial(2*n, n)/105: n in [0..30]]; // _Vincenzo Librandi_, Jul 05 2013

%o (PARI) vector(30, n, n--; m=n+4; binomial(m, 4)*binomial(2*m, m)/70) \\ _G. C. Greubel_, Jul 20 2019

%o (Sage) [binomial(n+4, 4)*binomial(2*(n+4), n+4)/70 for n in (0..30)] # _G. C. Greubel_, Jul 20 2019

%o (GAP) List([0..30], n-> Binomial(n+4, 4)*Binomial(2*(n+4), n+4)/70) # _G. C. Greubel_, Jul 20 2019

%Y Cf. A000302, A000332, A000984, A001622, A002457, A002697, A002802, A020918, A038845, A038846, A046521 (fifth column).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_