A084902 a(n) = 5^(n-1)*n*(n+1)/2.

0, 1, 15, 150, 1250, 9375, 65625, 437500, 2812500, 17578125, 107421875, 644531250, 3808593750, 22216796875, 128173828125, 732421875000, 4150390625000, 23345947265625, 130462646484375, 724792480468750, 4005432128906250

Offset: 0

Author

Paul Barry, Jun 10 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A084901. Fourth binomial transform of heptagonal numbers A000566. Fifth binomial transform of (0,1,5,0,0,0,...).

Number of n-permutations of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly two u's. Example: a(2)=15 because we have uuw, uuv, uuz, uux, uuy, uwu, uvu, uzu, uxu, uyu, wuu, vuu, zuu, xuu, yuu. - Zerinvary Lajos, Dec 30 2007

A shifted version of A081135. - R. J. Mathar, Sep 17 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (15,-75,125).

Crossrefs

Cf. A000566, A038243, A081135, A084901.

Programs

Magma

[5^(n-1)*Binomial(n+1,2): n in [0..30]]; // G. C. Greubel, May 17 2021

Mathematica

Table[5^(n-1)n(n+1)/2,{n,0,30}] (* or *) LinearRecurrence[{15,-75,125},{0,1,15},30] (* Harvey P. Dale, Sep 18 2018 *)

PARI

a(n)=5^(n-1)*n*(n+1)/2 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[5^(n-1)*binomial(n+1,2) for n in (0..30)] # G. C. Greubel, May 17 2021

Formula

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

E.g.f.: (x/2)*(2 + 5*x)*exp(5*x). - G. C. Greubel, May 17 2021

a(n) = 15*a(n-1) - 75*a(n-2) + 125*a(n-3). - Wesley Ivan Hurt, May 17 2021