A054338 8-fold convolution of A000302 (powers of 4).

1, 32, 576, 7680, 84480, 811008, 7028736, 56229888, 421724160, 2998927360, 20392706048, 133479530496, 845370359808, 5202279137280, 31213674823680, 183120225632256, 1052941297385472, 5946021444059136, 33033452466995200, 180814687187763200, 976399310813921280

Offset: 0

Author

Wolfdieter Lang, Mar 13 2000

Keywords

• easy
• nonn

Comments

With a different offset, number of n-permutations (n>=7) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly seven (7) u's. - Zerinvary Lajos, Jun 23 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (32,-448,3584,-17920,57344,-114688,131072,-65536).

Crossrefs

Cf. A000302, A054335.

Cf. A038231.

Programs

GAP

List([0..20], n-> 4^n*Binomial(n+7,7) ); # G. C. Greubel, Jul 21 2019

Magma

[4^n*Binomial(n+7, 7): n in [0..20]]; // Vincenzo Librandi, Oct 15 2011

Maple

seq(binomial(n+7,7)*4^n,n=0..20); # Zerinvary Lajos, Jun 23 2008

Mathematica

Table[4^n*Binomial[n+7,7], {n,0,20}] (* G. C. Greubel, Jul 21 2019 *)
LinearRecurrence[{32,-448,3584,-17920,57344,-114688,131072,-65536},{1,32,576,7680,84480,811008,7028736,56229888},30] (* Harvey P. Dale, Jun 08 2025 *)

PARI

vector(20, n, n--; 4^n*binomial(n+7,7)) \\ G. C. Greubel, Jul 21 2019

Formula

a(n) = binomial(n+7, 7)*4^n.

G.f.: 1/(1-4*x)^8.

a(n) = A054335(n+15, 15).

E.g.f.: (315 + 8820*x + 52920*x^2 + 117600*x^3 + 117600*x^4 + 56448*x^5 + 12544*x^6 + 1024*x^7)*exp(4*x)/315. - G. C. Greubel, Jul 21 2019

From Amiram Eldar, Mar 27 2022: (Start)

Sum_{n>=0} 1/a(n) = 20412*log(4/3) - 88067/15.

Sum_{n>=0} (-1)^n/a(n) = 437500*log(5/4) - 292873/3. (End)