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A036217 Expansion of 1/(1-3*x)^5; 5-fold convolution of A000244 (powers of 3).

%I A036217 #30 Sep 22 2022 03:58:40
%S A036217 1,15,135,945,5670,30618,153090,721710,3247695,14073345,59108049,
%T A036217 241805655,967222620,3794488740,14635885140,55616363532,208561363245,
%U A036217 772903875555,2833980877035,10291825290285,37050571045026
%N A036217 Expansion of 1/(1-3*x)^5; 5-fold convolution of A000244 (powers of 3).
%C A036217 With a different offset, number of n-permutations (n=5) of 4 objects: u, v, z, x with repetition allowed, containing exactly four (4) u's. Example: a(1)=15 because we have uuuuv uuuvu uuvuu uvuuu vuuuu uuuuz uuuzu uuzuu uzuuu zuuuu uuuux uuuxu uuxuu uxuuu xuuuu. - _Zerinvary Lajos_, Jun 12 2008
%H A036217 Vincenzo Librandi, <a href="/A036217/b036217.txt">Table of n, a(n) for n = 0..400</a>
%H A036217 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15,-90,270,-405,243).
%F A036217 a(n) = 3^n*binomial(n+4, 4) = 3^n*A000332(n+4).
%F A036217 a(n) = A027465(n+5, 5).
%F A036217 G.f.: 1/(1-3*x)^5.
%F A036217 E.g.f.: (1/8)*(8 +96*x +216*x^2 +144*x^3 +27*x^4)*exp(3*x). - _G. C. Greubel_, May 19 2021
%F A036217 From _Amiram Eldar_, Sep 22 2022: (Start)
%F A036217 Sum_{n>=0} 1/a(n) = 40 - 96*log(3/2).
%F A036217 Sum_{n>=0} (-1)^n/a(n) = 768*log(4/3) - 220. (End)
%p A036217 seq(3^n*binomial(n+4,4), n=0..30); # _Zerinvary Lajos_, Jun 12 2008
%t A036217 CoefficientList[Series[1/(1-3x)^5,{x,0,30}],x] (* _Harvey P. Dale_, Jun 13 2017 *)
%o A036217 (Sage) [3^n*binomial(n+4,4) for n in range(30)] # _Zerinvary Lajos_, Mar 10 2009
%o A036217 (Magma) [3^n* Binomial(n+4, 4): n in [0..30]]; // _Vincenzo Librandi_, Oct 14 2011
%Y A036217 Cf. A000332, A027465.
%Y A036217 Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), this sequence (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).
%K A036217 easy,nonn
%O A036217 0,2
%A A036217 _Wolfdieter Lang_