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 A036216 #43 Feb 16 2025 08:32:37
%S A036216 1,12,90,540,2835,13608,61236,262440,1082565,4330260,16888014,
%T A036216 64481508,241805655,892820880,3252418920,11708708112,41712272649,
%U A036216 147219785820,515269250370,1789882659180,6175095174171,21171754882872
%N A036216 Expansion of 1/(1 - 3*x)^4; 4-fold convolution of A000244 (powers of 3).
%C A036216 With three leading zeros, 3rd binomial transform of (0,0,0,1,0,0,0,0,...). - _Paul Barry_, Mar 07 2003
%C A036216 Number of n-permutations (n=4) of 4 objects u, v, w, z, with repetition allowed, containing exactly three u's. - _Zerinvary Lajos_, May 23 2008
%H A036216 Vincenzo Librandi, <a href="/A036216/b036216.txt">Table of n, a(n) for n = 0..400</a>
%H A036216 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>.
%H A036216 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-54,108,-81).
%F A036216 a(n) = 3^n*binomial(n+3, 3).
%F A036216 a(n) = A027465(n+4, 4).
%F A036216 G.f.: 1/(1 - 3*x)^4.
%F A036216 With three leading zeros, a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4), a(0) = a(1) = a(2) = 0, a(3) = 1. - _Paul Barry_, Mar 07 2003
%F A036216 With three leading zeros, C(n, 3)*3^(n-3) is the second binomial transform of C(n, 3). - _Paul Barry_, Jul 24 2003
%F A036216 E.g.f.: (1/2)*(2 + 18*x + 27*x^2 + 9*x^3)*exp(3*x). - _Franck Maminirina Ramaharo_, Nov 23 2018
%F A036216 From _Amiram Eldar_, Jan 05 2022: (Start)
%F A036216 Sum_{n>=0} 1/a(n) = 36*log(3/2) - 27/2.
%F A036216 Sum_{n>=0} (-1)^n/a(n) = 144*log(4/3) - 81/2. (End)
%p A036216 seq(3^n*binomial(n+3, 3), n=0..30)]; # _Zerinvary Lajos_, Dec 21 2006
%t A036216 CoefficientList[Series[1/(1-3x)^4,{x,0,30}],x] (* or *) LinearRecurrence[ {12,-54,108,-81},{1,12,90,540},30] (* _Harvey P. Dale_, Jul 27 2017 *)
%o A036216 (Sage) [3^n*binomial(n+3,3) for n in range(30)] # _Zerinvary Lajos_, Mar 10 2009
%o A036216 (Magma) [3^n* Binomial(n+3, 3): n in [0..30]]; // _Vincenzo Librandi_, Oct 14 2011
%o A036216 (PARI) a(n) = 3^n*binomial(n+3, 3) \\ _Charles R Greathouse IV_, Oct 03 2016
%Y A036216 Cf. A027465.
%Y A036216 Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), this sequence (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).
%K A036216 easy,nonn
%O A036216 0,2
%A A036216 _Wolfdieter Lang_