login
Expansion of 1/(1-3*x)^5; 5-fold convolution of A000244 (powers of 3).
16

%I #30 Sep 22 2022 03:58:40

%S 1,15,135,945,5670,30618,153090,721710,3247695,14073345,59108049,

%T 241805655,967222620,3794488740,14635885140,55616363532,208561363245,

%U 772903875555,2833980877035,10291825290285,37050571045026

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

%C 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 Vincenzo Librandi, <a href="/A036217/b036217.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15,-90,270,-405,243).

%F a(n) = 3^n*binomial(n+4, 4) = 3^n*A000332(n+4).

%F a(n) = A027465(n+5, 5).

%F G.f.: 1/(1-3*x)^5.

%F 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 From _Amiram Eldar_, Sep 22 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 40 - 96*log(3/2).

%F Sum_{n>=0} (-1)^n/a(n) = 768*log(4/3) - 220. (End)

%p seq(3^n*binomial(n+4,4), n=0..30); # _Zerinvary Lajos_, Jun 12 2008

%t CoefficientList[Series[1/(1-3x)^5,{x,0,30}],x] (* _Harvey P. Dale_, Jun 13 2017 *)

%o (Sage) [3^n*binomial(n+4,4) for n in range(30)] # _Zerinvary Lajos_, Mar 10 2009

%o (Magma) [3^n* Binomial(n+4, 4): n in [0..30]]; // _Vincenzo Librandi_, Oct 14 2011

%Y Cf. A000332, A027465.

%Y 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 easy,nonn

%O 0,2

%A _Wolfdieter Lang_