A081136 - OEIS

%I #39 Sep 08 2022 08:45:09

%S 0,0,1,18,216,2160,19440,163296,1306368,10077696,75582720,554273280,

%T 3990767616,28298170368,198087192576,1371372871680,9403699691520,

%U 63945157902336,431629815840768,2894458765049856,19296391766999040

%N 6th binomial transform of (0,0,1,0,0,0, ...).

%C Starting at 1, three-fold convolution of A000400 (powers of 6).

%C Number of n-permutations of 7 objects: p, u, v, w, z, x, y with repetition allowed, containing exactly two u's. - _Zerinvary Lajos_, May 23 2008

%H Vincenzo Librandi, <a href="/A081136/b081136.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (18,-108,216).

%F a(n) = 18*a(n-1) -108*a(n-2) +216*a(n-3), a(0)=a(1)=0, a(2)=1.

%F a(n) = 6^(n-2)*C(n, 2).

%F G.f.: x^2/(1-6*x)^3.

%F E.g.f.: exp(6*x) * x^2/2. - _Geoffrey Critzer_, Oct 03 2013

%F From _Amiram Eldar_, Jan 05 2022: (Start)

%F Sum_{n>=2} 1/a(n) = 12 - 60*log(6/5).

%F Sum_{n>=2} (-1)^n/a(n) = 84*log(7/6) - 12. (End)

%p seq(binomial(n, 2)*6^(n-2), n=0..19); # _Zerinvary Lajos_, May 23 2008

%t nn=20;Range[0,nn]!CoefficientList[Series[x^2/2! Exp[6x],{x,0,nn}],x] (* _Geoffrey Critzer_, Oct 03 2013 *)

%t LinearRecurrence[{18,-108,216},{0,0,1},30] (* _Harvey P. Dale_, Apr 20 2022 *)

%o (Sage) [6^(n-2)*binomial(n,2) for n in range(0, 21)] # _Zerinvary Lajos_, Mar 13 2009

%o (Magma) [6^n*Binomial(n+2,2): n in [-2..20]]; // _Vincenzo Librandi_, Oct 16 2011

%Y Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), this sequence (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

%K easy,nonn

%O 0,4

%A _Paul Barry_, Mar 08 2003