A020920 Expansion of 1/(1-4*x)^(9/2).

1, 18, 198, 1716, 12870, 87516, 554268, 3325608, 19122246, 106234700, 573667380, 3024791640, 15628090140, 79342611480, 396713057400, 1957117749840, 9540949030470, 46021048264620, 219878341708740, 1041528987041400, 4895186239094580, 22844202449108040

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

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

Programs

GAP

List([0..30], n-> Binomial(n+4, 4)*Binomial(2*(n+4), n+4)/70) # G. C. Greubel, Jul 20 2019

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

Maple

seq(binomial(2*n+8, n+4)*binomial(n+4, n)/70, n=0..30); # Zerinvary Lajos, May 05 2007

Mathematica

CoefficientList[Series[1/(1-4x)^(9/2), {x,0,30}], x] (* Vincenzo Librandi, Jul 05 2013 *)

PARI

vector(30, n, n--; m=n+4; binomial(m, 4)*binomial(2*m, m)/70) \\ G. C. Greubel, Jul 20 2019

Sage

[binomial(n+4, 4)*binomial(2*(n+4), n+4)/70 for n in (0..30)] # G. C. Greubel, Jul 20 2019

Formula

a(n) = binomial(n+4, 4)*A000984(n+4)/A000984(4), where A000984 are the central binomial coefficients. - Wolfdieter Lang

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

a(n) = A000332(n+4)*A000984(n+4)/70. - Zerinvary Lajos, May 05 2007

From Rui Duarte, Oct 08 2011: (Start)

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

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

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

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.

From Amiram Eldar, Mar 25 2022: (Start)

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

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