A020922 Expansion of 1/(1-4*x)^(11/2).

1, 22, 286, 2860, 24310, 184756, 1293292, 8498776, 53117350, 318704100, 1848483780, 10418726760, 57302997180, 308554600200, 1630931458200, 8480843582640, 43464323361030, 219878341708740, 1099391708543700, 5439095821216200, 26651569523959380, 129450480544945560

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also convolution of A000984 with A040075, also convolution of A000302 with A020920, also convolution of A002457 with A038846, also convolution of A002697 with A020918, also convolution of A002802 with A038845. - Rui Duarte, Oct 08 2011

Links

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

Crossrefs

Cf. A000302, A000984, A001622, A002457, A002697, A002802, A020918, A020920, A038845, A038846, A040075, A046521 (sixth column).

Programs

GAP

List([0..30], n-> Binomial(n+5, 5)*Binomial(2*n+10, n+5)/252); # G. C. Greubel, Jul 20 2019

Magma

[(2*n+9)*(2*n+7)*(2*n+5)*(2*n+3)*(2*n+1)*Binomial(2*n, n)/945: n in [0..30]] // Vincenzo Librandi, Jul 05 2013

Mathematica

CoefficientList[Series[1/(1-4x)^(11/2), {x,0,30}], x] (* Vincenzo Librandi, Jul 05 2013 *)

PARI

vector(30, n, n--; m=n+5; binomial(m, 5)*binomial(2*m, m)/252) \\ G. C. Greubel, Jul 20 2019

Sage

[binomial(n+5, 5)*binomial(2*n+10, n+5)/252 for n in (0..30)] # G. C. Greubel, Jul 20 2019

Formula

a(n) = binomial(n+5, 5)*A000984(n+5)/A000984(5), where A000984 are central binomial coefficients. - Wolfdieter Lang

From Rui Duarte, Oct 08 2011: (Start)

a(n) = ((2n+9)(2n+7)(2n+5)(2n+3)(2n+1)/(9*7*5*3*1)) * binomial(2n, n).

a(n) = binomial(2n+10, 10) * binomial(2n, n) / binomial(n+5, 5).

a(n) = binomial(n+5, 5) * binomial(2n+10, n+5) / binomial(10, 5).

a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11 = n } f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10)*f(i_11) with f(k)=A000984(k). (End)

Boas-Buck recurrence: a(n) = (22/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+5, 5). See a comment there. - Wolfdieter Lang, Aug 10 2017

From Amiram Eldar, Mar 25 2022: (Start)

Sum_{n>=0} 1/a(n) = 162*sqrt(3)*Pi - 30816/35.

Sum_{n>=0} (-1)^n/a(n) = 4500*sqrt(5)*log(phi) - 33888/7, where phi is the golden ratio (A001622). (End)