A081141 11th binomial transform of (0,0,1,0,0,0,...).

0, 0, 1, 33, 726, 13310, 219615, 3382071, 49603708, 701538156, 9646149645, 129687123005, 1711870023666, 22254310307658, 285596982281611, 3624884775112755, 45569980029988920, 568105751040528536

Offset: 0

Author

Paul Barry, Mar 08 2003

Keywords

• easy
• nonn

Comments

Starting at 1, the three-fold convolution of A001020 (powers of 11).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (33,-363,1331).

Crossrefs

Cf. A001020.

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), this sequence (q=11), A081142 (q=12), A027476 (q=15).

Programs

Magma

[11^(n-2)*Binomial(n, 2): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Maple

seq((11)^(n-2)*binomial(n,2), n=0..30); # G. C. Greubel, May 13 2021

Mathematica

LinearRecurrence[{33,-363,1331},{0,0,1},30] (* Harvey P. Dale, Dec 15 2014 *)

PARI

vector(20, n, n--; 11^(n-2)*binomial(n, 2)) \\ G. C. Greubel, Nov 23 2018

Sage

[11^(n-2)*binomial(n, 2) for n in range(20)] # G. C. Greubel, Nov 23 2018

Formula

a(n) = 33*a(n-1) - 363*a(n-2) + 1331*a(n-3), a(0) = a(1) = 0, a(2) = 1.

a(n) = 11^(n-2)*binomial(n, 2).

G.f.: x^2/(1 - 11*x)^3.

E.g.f.: (1/2)*exp(11*x)*x^2. - Franck Maminirina Ramaharo, Nov 23 2018

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=2} 1/a(n) = 22 - 220*log(11/10).

Sum_{n>=2} (-1)^n/a(n) = 264*log(12/11) - 22. (End)