This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A036223 #30 Sep 22 2022 01:56:01
%S A036223 1,30,495,5940,57915,486486,3648645,25019280,159497910,956987460,
%T A036223 5454828522,29753610120,156206453130,793048146660,3908594437110,
%U A036223 18761253298128,87943374834975,403504896301650,1815772033357425
%N A036223 Expansion of 1/(1-3*x)^10; 10-fold convolution of A000244 (powers of 3).
%C A036223 With a different offset, number of n-permutations (n >= 9) of 4 objects: u, v, z, x with repetition allowed, containing exactly nine (9) u's. - _Zerinvary Lajos_, Jul 02 2008
%H A036223 Vincenzo Librandi, <a href="/A036223/b036223.txt">Table of n, a(n) for n = 0..400</a>
%H A036223 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049).
%F A036223 a(n) = 3^n*binomial(n+9, 9).
%F A036223 a(n) = A027465(n+10, 10).
%F A036223 G.f.: 1/(1-3*x)^10.
%F A036223 E.g.f.: (4480 + 120960*x + 725760*x^2 + 1693440*x^3 + 1905120*x^4 + 1143072*x^5 + 381024*x^6 + 69984*x^7 + 6561*x^8 + 243*x^9)*exp(3*x)/4480. - _G. C. Greubel_, May 18 2021
%F A036223 From _Amiram Eldar_, Sep 22 2022: (Start)
%F A036223 Sum_{n>=0} 1/a(n) = 6912*log(3/2) - 784431/280.
%F A036223 Sum_{n>=0} (-1)^n/a(n) = 1769472*log(4/3) - 142532433/280. (End)
%p A036223 seq(3^n*binomial(n+9, 9), n=0..20); # _Zerinvary Lajos_, Jul 02 2008
%t A036223 Table[3^n*Binomial[n+9,9], {n,0,30}] (* _G. C. Greubel_, May 18 2021 *)
%t A036223 CoefficientList[Series[1/(1-3x)^10,{x,0,30}],x] (* or *) LinearRecurrence[ {30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049},{1,30,495,5940,57915,486486,3648645,25019280,159497910,956987460},30] (* _Harvey P. Dale_, Jan 16 2022 *)
%o A036223 (Sage) [3^n*binomial(n+9,9) for n in range(30)] # _Zerinvary Lajos_, Mar 13 2009
%o A036223 (Magma) [3^n*Binomial(n+9, 9): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011
%Y A036223 Cf. A027465.
%Y A036223 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), A036221 (m=7), A036222 (m=8), this sequence (m=9), A172362 (m=10).
%K A036223 easy,nonn
%O A036223 0,2
%A A036223 _Wolfdieter Lang_