A027474 a(n) = 7^(n-2) * C(n,2).

1, 21, 294, 3430, 36015, 352947, 3294172, 29647548, 259416045, 2219448385, 18643366434, 154231485954, 1259557135291, 10173346092735, 81386768741880, 645668365352248, 5084638377148953, 39779817891812397, 309398583602985310

Offset: 2

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

7th binomial transform of (0,0,1,0,0,0,........). Starting at 1, the three-fold convolution of A000420 (powers of 7). - Paul Barry, Mar 08 2003

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..400
• Index entries for linear recurrences with constant coefficients, signature (21,-147,343).

Crossrefs

Third column of A027466.

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), this sequence (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

Programs

Magma

[7^(n-2)* Binomial(n, 2): n in [2..20]]; /* Vincenzo Librandi, Oct 12 2011 */

Maple

seq(binomial(n, 2)*7^(n-2), n=2..30); # Zerinvary Lajos, Jun 12 2008

Mathematica

Table[7^(n-2) Binomial[n,2], {n,2,20}] (* Harvey P. Dale, Sep 25 2011 *)

PARI

a(n)=7^(n-2)*n*(n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

Sage

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

Formula

From Paul Barry, Mar 08 2003: (Start)

G.f.: x^2 / (1-7*x)^3.

a(n) = 21*a(n-1) - 147*a(n-2) + 343*a(n-3), a(0) = a(1) = 0, a(2) = 1. (End)

Numerators of sequence a[3,n] in (a[i,j])^3 where a[i,j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.

E.g.f.: (x^2/2)*exp(7*x). - G. C. Greubel, May 13 2021

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=2} 1/a(n) = 14 - 84*log(7/6).

Sum_{n>=2} (-1)^n/a(n) = 112*log(8/7) - 14. (End)

Extensions

Edited by Ralf Stephan, Dec 30 2004