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%I #37 Mar 25 2022 09:14:46
%S 1,24,336,3584,32256,258048,1892352,12976128,84344832,524812288,
%T 3148873728,18320719872,103817412608,574988746752,3121367482368,
%U 16647293239296,87398289506304,452414675091456,2312341672689664,11683410556747776,58417052783738880,289303499500421120
%N 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.
%C Also convolution of A020922 with A000984 (central binomial coefficients); also convolution of A040075 with A000302 (powers of 4).
%C With a different offset, number of n-permutations of 5 objects: u,v,z,x, y with repetition allowed, containing exactly five (5) u's. Example: a(1)=24 because we have uuuuuv uuuuvu uuuvuu uuvuuu uvuuuu vuuuuu uuuuuz uuuuzu uuuzuu uuzuuu uzuuuu zuuuuu uuuuux uuuuxu uuuxuu uuxuuu uxuuuu xuuuuu uuuuuy uuuuyu uuuyuu uuyuuu uyuuuu yuuuuu. - _Zerinvary Lajos_, Jun 16 2008
%C Also convolution of A002457 with A020920, also convolution of A002697 with A038846, also convolution of A002802 with A020918, also convolution of A038845 with A038845. - _Rui Duarte_, Oct 08 2011
%H Vincenzo Librandi, <a href="/A045543/b045543.txt">Table of n, a(n) for n = 0..400</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (24,-240,1280,-3840,6144,-4096).
%F a(n) = binomial(n+5, 5)*4^n.
%F G.f.: 1/(1-4*x)^6.
%F a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11+i_12 = n} f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) *f(i_11)*f(i_12), with f(k)=A000984(k). - _Rui Duarte_, Oct 08 2011
%F E.g.f.: (15 + 120*x + 240*x^2 + 160*x^3 + 32*x^4)*exp(4*x)/3. - _G. C. Greubel_, Jul 20 2019
%F From _Amiram Eldar_, Mar 25 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 1620*log(4/3) - 465.
%F Sum_{n>=0} (-1)^n/a(n) = 12500*log(5/4) - 8365/3. (End)
%p seq(seq(binomial(i+5, j)*4^i, j =i), i=0..30); # _Zerinvary Lajos_, Dec 03 2007
%p seq(binomial(n+5,5)*4^n,n=0..30); # _Zerinvary Lajos_, Jun 16 2008
%t CoefficientList[Series[1/(1-4x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {24,-240,1280,-3840,6144,-4096}, {1,24,336,3584,32256, 258048}, 30] (* _Harvey P. Dale_, Mar 24 2018 *)
%o (Sage) [lucas_number2(n, 4, 0)*binomial(n,5)/2^10 for n in range(5, 35)] # _Zerinvary Lajos_, Mar 11 2009
%o (Magma) [4^n*Binomial(n+5, 5): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011
%o (PARI) Vec(1/(1-4*x)^6 + O(x^30)) \\ _Michel Marcus_, Aug 21 2015
%o (GAP) List([0..30], n-> 4^n*Binomial(n+5,5)); # _G. C. Greubel_, Jul 20 2019
%Y Cf. A038231.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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