A017389 Expansion of g.f. 1/((1-3*x)*(1-5*x)*(1-7*x)).

1, 15, 154, 1350, 10891, 83685, 623764, 4558380, 32875381, 234980955, 1669192174, 11806149810, 83252603071, 585817587825, 4115974729384, 28888095527640, 202598073849961, 1420093671872295, 9950191865139394, 69699025028403870, 488131588547752051, 3418113197039242365

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (15,-71,105).

Crossrefs

Cf. A019333.

Programs

Magma

I:=[1,15,154]; [n le 3 select I[n] else 15*Self(n-1)-71*Self(n-2)+105*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 26 2013

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-5*x)*(1-7*x)))); // Vincenzo Librandi, Jun 26 2013

Maple

A017389:=n->(7^(n+2) - 2*5^(n+2) + 3^(n+2))/8: seq(A017389(n), n=0..20); # Wesley Ivan Hurt, Mar 25 2014

Mathematica

CoefficientList[Series[1 / ((1 - 3 x) (1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 26 2013 *)

PARI

a(n) = (7^(n+2) - 2*5^(n+2) + 3^(n+2))/8; \\ Joerg Arndt, Aug 13 2013

PARI

x='x+O('x^20); Vec(1/((1-3*x)*(1-5*x)*(1-7*x))) \\ Altug Alkan, Sep 23 2018

Formula

From Vincenzo Librandi, Jun 26 2013: (Start)

a(n) = 15*a(n-1) - 71*a(n-2) + 105*a(n-3).

a(n) = 12*a(n-1) - 35*a(n-2) + 3^n. (End)

a(n) = (7^(n+2) - 2*5^(n+2) + 3^(n+2))/8. - Yahia Kahloune, Aug 13 2013

From Seiichi Manyama, May 04 2025: (Start)

a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).

a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

E.g.f.: exp(3*x)*(9 - 50*exp(2*x) + 49*exp(4*x))/8. - Stefano Spezia, May 04 2025