A054339 9-fold convolution of A000302 (powers of 4).

1, 36, 720, 10560, 126720, 1317888, 12300288, 105431040, 843448320, 6372720640, 45883588608, 317013884928, 2113425899520, 13655982735360, 85837605765120, 526470648692736, 3158823892156416, 18581317012684800, 107358720517734400, 610249569258700800

Offset: 0

Author

Wolfdieter Lang, Mar 13 2000

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Index entries for linear recurrences with constant coefficients, signature (36,-576,5376,-32256,129024,-344064,589824,-589824,262144).

Crossrefs

Cf. A000302, A054335.

Cf. A038231.

Programs

GAP

List([0..20], n-> 4^n*Binomial(n+8, 8)); # G. C. Greubel, Jul 21 2019

Magma

[Binomial(n+8, 8)*4^n: n in [0..20]]; // Vincenzo Librandi, May 31 2011

Maple

seq(binomial(n+8,8)*4^n,n=0..20); # Zerinvary Lajos, Jun 23 2008

Mathematica

Table[Binomial[n+8,8]4^n,{n,0,20}] (* or *) LinearRecurrence[ {36,-576,5376,-32256,129024,-344064,589824,-589824,262144},{1,36,720,10560,126720,1317888,12300288,105431040,843448320},20]

PARI

vector(20, n, n--; 4^n*binomial(n+8, 8)) \\ G. C. Greubel, Jul 21 2019

Sage

[4^n*binomial(n+8, 8) for n in (0..20)] # G. C. Greubel, Jul 21 2019

Formula

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

G.f.: 1/(1-4*x)^9.

a(n) = A054335(n+17, 17).

a(n) = 36*a(n-1) - 576*a(n-2) + 5376*a(n-3) - 32256*a(n-4) + 129024*a(n-5) - 344064*a(n-6) + 589824*a(n-7) - 589824*a(n-8) + 262144*a(n-9). - Harvey P. Dale, Aug 30 2013

E.g.f.: (16/7!)*(315 + 10080*x + 70560*x^2 + 188160*x^3 + 235200*x^4 + 150528*x^5 + 50176*x^6 + 8192*x^7 + 512*x^8)*exp(4*x). - G. C. Greubel, Jul 21 2019

From Amiram Eldar, Mar 27 2022: (Start)

Sum_{n>=0} 1/a(n) = 704696/35 - 69984*log(4/3).

Sum_{n>=0} (-1)^n/a(n) = 2500000*log(5/4) - 11715016/21. (End)