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

0,2

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).

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)

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]

Prog

(Magma) [Binomial(n+8, 8)*4^n: n in [0..20]]; // Vincenzo Librandi, May 31 2011
(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
(GAP) List([0..20], n-> 4^n*Binomial(n+8, 8)); # G. C. Greubel, Jul 21 2019

Crossrefs

Cf. A000302, A054335.

Cf. A038231.

Sequence in context: A268900 A185491 A220758 * A232131 A138832 A223998

Adjacent sequences: A054336 A054337 A054338 * A054340 A054341 A054342

Keyword

easy,nonn

Author

Wolfdieter Lang, Mar 13 2000

Status

approved