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

0,4

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).

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)

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 *)

Prog

(Sage) [6^(n-2)*binomial(n, 2) for n in range(0, 21)] # Zerinvary Lajos, Mar 13 2009
(Magma) [6^n*Binomial(n+2, 2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

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).

Sequence in context: A324638 A009470 A111991 * A101188 A019757 A021503

Adjacent sequences: A081133 A081134 A081135 * A081137 A081138 A081139

Keyword

easy,nonn

Author

Paul Barry, Mar 08 2003

Status

approved