A081136 6th binomial transform of (0,0,1,0,0,0, ...).

0, 0, 1, 18, 216, 2160, 19440, 163296, 1306368, 10077696, 75582720, 554273280, 3990767616, 28298170368, 198087192576, 1371372871680, 9403699691520, 63945157902336, 431629815840768, 2894458765049856, 19296391766999040

Offset: 0

Author

Paul Barry, Mar 08 2003

Keywords

• easy
• nonn

Comments

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (18,-108,216).

Crossrefs

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).

Programs

Magma

[6^n*Binomial(n+2,2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

Maple

seq(binomial(n, 2)*6^(n-2), n=0..19); # Zerinvary Lajos, May 23 2008

Mathematica

nn=20;Range[0,nn]!CoefficientList[Series[x^2/2! Exp[6x],{x,0,nn}],x] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{18,-108,216},{0,0,1},30] (* Harvey P. Dale, Apr 20 2022 *)

Sage

[6^(n-2)*binomial(n,2) for n in range(0, 21)] # Zerinvary Lajos, Mar 13 2009

Formula

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

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

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

E.g.f.: exp(6*x) * x^2/2. - Geoffrey Critzer, Oct 03 2013

From Amiram Eldar, Jan 05 2022: (Start)

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

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