A054337 7-fold convolution of A000302 (powers of 4).

1, 28, 448, 5376, 53760, 473088, 3784704, 28114944, 196804608, 1312030720, 8396996608, 51908706304, 311452237824, 1820797698048, 10404558274560, 58265526337536, 320460394856448, 1734256254517248, 9249366690758656, 48680877319782400, 253140562062868480

Offset: 0

Author

Wolfdieter Lang, Mar 13 2000

Keywords

• easy
• nonn

Comments

With a different offset, number of n-permutations (n>=6) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly six (6) u's. Example: a(1)=28 because we have uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu, uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu, uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu, uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu. - Zerinvary Lajos, Jun 16 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A000302, A054335.

Cf. A038231.

Programs

GAP

List([0..30], n-> 4^n*Binomial(n+6,6)); # G. C. Greubel, Jul 21 2019

Magma

[4^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple

seq(seq(binomial(i, j)*4^(i-6), j =i-6), i=6..36); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+6,6)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008

Mathematica

Table[4^n*Binomial[n+6,6], {n,0,30}] (* G. C. Greubel, Jul 21 2019 *)

PARI

vector(30, n, n--; 4^n*binomial(n+6,6) ) \\ G. C. Greubel, Jul 21 2019

Sage

[lucas_number2(n, 4, 0)*binomial(n,6)/2^12 for n in range(6, 36)] # Zerinvary Lajos, Mar 11 2009

Formula

a(n) = binomial(n+6, 6)*4^n.

G.f.: 1/(1 - 4*x)^7.

a(n) = A054335(n+13, 13).

E.g.f.: (45 + 1080*x + 5400*x^2 + 9600*x^3 + 7200*x^4 + 2304*x^5 + 256*x^6)*exp(4*x)/45. - G. C. Greubel, Jul 21 2019

From Amiram Eldar, Mar 25 2022: (Start)

Sum_{n>=0} 1/a(n) = 8394/5 - 5832*log(4/3).

Sum_{n>=0} (-1)^n/a(n) = 75000*log(5/4) - 83674/5. (End)