A081142 12th binomial transform of (0,0,1,0,0,0,...).

0, 0, 1, 36, 864, 17280, 311040, 5225472, 83607552, 1289945088, 19349176320, 283787919360, 4086546038784, 57954652913664, 811365140791296, 11234286564802560, 154070215745863680, 2095354934143746048

Offset

0,4

Comments

Starting at 1, the three-fold convolution of A001021 (powers of 12).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (36,-432,1728).

Formula

a(n) = 36*a(n-1) - 432*a(n-2) + 1728*a(n-3), a(0) = a(1) = 0, a(2) = 1.

a(n) = 12^(n-2)*binomial(n, 2).

G.f.: x^2/(1 - 12*x)^3.

a(n) = 2^(2*n-5)*3^(n-2)*n*(n-1). - Harvey P. Dale, Jul 25 2013

E.g.f.: (1/2)*exp(12*x)*x^2. - Franck Maminirina Ramaharo, Nov 23 2018

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=2} 1/a(n) = 24 - 264*log(12/11).

Sum_{n>=2} (-1)^n/a(n) = 312*log(13/12) - 24. (End)

Maple

seq(coeff(series(x^2/(1-12*x)^3, x, n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Nov 24 2018

Mathematica

LinearRecurrence[{36, -432, 1728}, {0, 0, 1}, 30] (* or *) Table[(n-1) (n-2) 3^(n-3) 2^(2n-7), {n, 20}] (* Harvey P. Dale, Jul 25 2013 *)

Prog

(Magma) [12^(n-2)* Binomial(n, 2): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011
(PARI) vector(20, n, n--; 2^(2*n-5)*3^(n-2)*n*(n-1)) \\ G. C. Greubel, Nov 23 2018
(Sage) [2^(2*n-5)*3^(n-2)*n*(n-1) for n in range(20)] # G. C. Greubel, Nov 23 2018
(GAP) List([0..20], n->12^(n-2)*Binomial(n, 2)); # Muniru A Asiru, Nov 24 2018

Crossrefs

Cf. A001021.

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), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), this sequence (q=12), A027476 (q=15).

Sequence in context: A203271 A004360 A238931 * A061694 A264192 A122038

Adjacent sequences: A081139 A081140 A081141 * A081143 A081144 A081145

Keyword

easy,nonn

Author

Paul Barry, Mar 08 2003

Status

approved