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A036219 Expansion of 1/(1-3*x)^6; 6-fold convolution of A000244 (powers of 3).

%I A036219 #35 Sep 22 2022 03:58:26
%S A036219 1,18,189,1512,10206,61236,336798,1732104,8444007,39405366,177324147,
%T A036219 773778096,3288556908,13660159464,55616363532,222465454128,
%U A036219 875957725629,3400777052442,13036312034361,49400761393368,185252855225130,688082033693340,2533392942234570
%N A036219 Expansion of 1/(1-3*x)^6; 6-fold convolution of A000244 (powers of 3).
%H A036219 Vincenzo Librandi, <a href="/A036219/b036219.txt">Table of n, a(n) for n = 0..400</a>
%H A036219 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (18,-135,540,-1215,1458,-729).
%F A036219 a(n) = 3^n*binomial(n+5, 5).
%F A036219 a(n) = A027465(n+6, 6).
%F A036219 G.f.: 1/(1-3*x)^6.
%F A036219 E.g.f.: (1/40)*(40 + 600*x + 1800*x^2 + 1800*x^3 + 675*x^4 + 81*x^5)*exp(3*x). - _G. C. Greubel_, May 19 2021
%F A036219 From _Amiram Eldar_, Sep 22 2022: (Start)
%F A036219 Sum_{n>=0} 1/a(n) = 240*log(3/2) - 385/4.
%F A036219 Sum_{n>=0} (-1)^n/a(n) = 3840*log(4/3) - 4415/4. (End)
%p A036219 seq(3^n*binomial(n+5,5), n=0..30); # _Zerinvary Lajos_, Jun 13 2008
%t A036219 Table[3^n*Binomial[n+5, 5], {n, 0, 30}] (* _G. C. Greubel_, May 19 2021 *)
%t A036219 CoefficientList[Series[1/(1-3x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {18,-135,540,-1215,1458,-729},{1,18,189,1512,10206,61236},30] (* _Harvey P. Dale_, Jan 02 2022 *)
%o A036219 (Sage) [3^n*binomial(n+5,5) for n in range(30)] # _Zerinvary Lajos_, Mar 10 2009
%o A036219 (Magma) [3^n*Binomial(n+5, 5): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011
%Y A036219 Cf. A027465.
%Y A036219 Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), this sequence (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).
%K A036219 easy,nonn
%O A036219 0,2
%A A036219 _Wolfdieter Lang_