cp's OEIS Frontend

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

%I A054337 #25 Mar 25 2022 09:14:51
%S A054337 1,28,448,5376,53760,473088,3784704,28114944,196804608,1312030720,
%T A054337 8396996608,51908706304,311452237824,1820797698048,10404558274560,
%U A054337 58265526337536,320460394856448,1734256254517248,9249366690758656,48680877319782400,253140562062868480
%N A054337 7-fold convolution of A000302 (powers of 4).
%C A054337 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
%H A054337 Vincenzo Librandi, <a href="/A054337/b054337.txt">Table of n, a(n) for n = 0..400</a>
%F A054337 a(n) = binomial(n+6, 6)*4^n.
%F A054337 G.f.: 1/(1 - 4*x)^7.
%F A054337 a(n) = A054335(n+13, 13).
%F A054337 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
%F A054337 From _Amiram Eldar_, Mar 25 2022: (Start)
%F A054337 Sum_{n>=0} 1/a(n) = 8394/5 - 5832*log(4/3).
%F A054337 Sum_{n>=0} (-1)^n/a(n) = 75000*log(5/4) - 83674/5. (End)
%p A054337 seq(seq(binomial(i, j)*4^(i-6), j =i-6), i=6..36); # _Zerinvary Lajos_, Dec 03 2007
%p A054337 seq(binomial(n+6,6)*4^n,n=0..30); # _Zerinvary Lajos_, Jun 16 2008
%t A054337 Table[4^n*Binomial[n+6,6], {n,0,30}] (* _G. C. Greubel_, Jul 21 2019 *)
%o A054337 (Sage) [lucas_number2(n, 4, 0)*binomial(n,6)/2^12 for n in range(6, 36)] # _Zerinvary Lajos_, Mar 11 2009
%o A054337 (Magma) [4^n*Binomial(n+6, 6): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011
%o A054337 (PARI) vector(30, n, n--; 4^n*binomial(n+6,6) ) \\ _G. C. Greubel_, Jul 21 2019
%o A054337 (GAP) List([0..30], n-> 4^n*Binomial(n+6,6)); # _G. C. Greubel_, Jul 21 2019
%Y A054337 Cf. A000302, A054335.
%Y A054337 Cf. A038231.
%K A054337 easy,nonn
%O A054337 0,2
%A A054337 _Wolfdieter Lang_, Mar 13 2000