A081135 5th binomial transform of (0,0,1,0,0,0, ...).

0, 0, 1, 15, 150, 1250, 9375, 65625, 437500, 2812500, 17578125, 107421875, 644531250, 3808593750, 22216796875, 128173828125, 732421875000, 4150390625000, 23345947265625, 130462646484375, 724792480468750

Offset: 0

Author

Paul Barry, Mar 08 2003

Keywords

• easy
• nonn

Comments

Starting at 1, three-fold convolution of A000351 (powers of 5).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (15,-75,125).

Crossrefs

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), this sequence (q=5), A081136 (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

[5^(n-2)*Binomial(n, 2): n in [0..30]]; // Vincenzo Librandi, Aug 06 2013

Maple

seq(n*(n-1)*5^(n-2)/2, n=0..30); # Zerinvary Lajos, May 03 2007

Mathematica

CoefficientList[Series[x^2/(1-5x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{15,-75,125},{0,0,1},30] (* Harvey P. Dale, Sep 13 2017 *)

Sage

[5^(n-2)*binomial(n,2) for n in range(0, 30)] # Zerinvary Lajos, Mar 12 2009

Formula

a(n) = 15*a(n-1) - 75*a(n-2) + 125*a(n-3), a(0)=a(1)=0, a(2)=1.

a(n) = 5^(n-2)*binomial(n, 2).

G.f.: x^2/(1-5*x)^3.

E.g.f.: (x^2/2)*exp(5*x). - G. C. Greubel, May 14 2021

From Amiram Eldar, Jan 05 2022: (Start)

Sum_{n>=2} 1/a(n) = 10 - 40*log(5/4).

Sum_{n>=2} (-1)^n/a(n) = 60*log(6/5) - 10. (End)