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

0,2

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

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)

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 *)

Prog

(Sage) [lucas_number2(n, 4, 0)*binomial(n, 6)/2^12 for n in range(6, 36)] # Zerinvary Lajos, Mar 11 2009
(Magma) [4^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011
(PARI) vector(30, n, n--; 4^n*binomial(n+6, 6) ) \\ G. C. Greubel, Jul 21 2019
(GAP) List([0..30], n-> 4^n*Binomial(n+6, 6)); # G. C. Greubel, Jul 21 2019

Crossrefs

Cf. A000302, A054335.

Cf. A038231.

Sequence in context: A022656 A154588 A125485 * A009685 A007833 A080315

Adjacent sequences: A054334 A054335 A054336 * A054338 A054339 A054340

Keyword

easy,nonn

Author

Wolfdieter Lang, Mar 13 2000

Status

approved