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A036221 Expansion of 1/(1-3*x)^8; 8-fold convolution of A000244 (powers of 3).

%I A036221 #27 Sep 22 2022 01:55:04
%S A036221 1,24,324,3240,26730,192456,1250964,7505784,42220035,225173520,
%T A036221 1148384952,5637526128,26778249108,123591918960,556163635320,
%U A036221 2447119995408,10553204980197,44695926974952,186233029062300
%N A036221 Expansion of 1/(1-3*x)^8; 8-fold convolution of A000244 (powers of 3).
%C A036221 With a different offset, number of n-permutations (n>=7) of 4 objects: u, v, z, x with repetition allowed, containing exactly seven (7) u's. Example: a(1)=24 because we have uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu. - _Zerinvary Lajos_, Jun 23 2008
%H A036221 Vincenzo Librandi, <a href="/A036221/b036221.txt">Table of n, a(n) for n = 0..400</a>
%H A036221 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (24,-252,1512,-5670,13608,-20412,17496,-6561).
%F A036221 a(n) = 3^n*binomial(n+7, 7).
%F A036221 a(n) = A027465(n+8, 8.)
%F A036221 G.f.: 1/(1-3*x)^8.
%F A036221 E.g.f.: (1/560)*(560 +11760*x +52920*x^2 +88200*x^3 +66150*x^4 +23814*x^5 +3969*x^6 +243*x^7)*exp(3*x). - _G. C. Greubel_, May 19 2021
%F A036221 From _Amiram Eldar_, Sep 22 2022: (Start)
%F A036221 Sum_{n>=0} 1/a(n) = 1344*log(3/2) - 5439/10.
%F A036221 Sum_{n>=0} (-1)^n/a(n) = 86016*log(4/3) - 247443/10. (End)
%p A036221 seq(3^n*binomial(n+7,7), n=0..30); # _Zerinvary Lajos_, Jun 23 2008
%t A036221 Table[3^n*Binomial[n+7,7], {n,0,30}] (* _G. C. Greubel_, May 19 2021 *)
%o A036221 (Sage) [3^n*binomial(n+7, 7) for n in range(30)] # _Zerinvary Lajos_, Mar 13 2009
%o A036221 (Magma) [3^n*Binomial(n+7, 7): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011
%Y A036221 Cf. A027465.
%Y A036221 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), A036219 (m=5), A036220 (m=6), this sequence (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).
%K A036221 easy,nonn
%O A036221 0,2
%A A036221 _Wolfdieter Lang_