A016315 Expansion of g.f. 1/((1 - 2*x)*(1 - 7*x)*(1 - 12*x)).

1, 21, 319, 4305, 55015, 683697, 8369047, 101581473, 1227048295, 14781074385, 177768357559, 2135988547329, 25651240368391, 307950529031985, 3696355860679255, 44362916914251873, 532401529073793703, 6389144031605054097, 76672008158297618935, 920080056352830739905

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 (21,-122,168).

Crossrefs

Cf. A016307.

Programs

Magma

I:=[1, 21, 319]; [n le 3 select I[n] else 21*Self(n-1)-122*Self(n-2)+168*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 26 2013

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1/((1-2*x)*(1-7*x)*(1-12*x))))); // Vincenzo Librandi, Jun 26 2013

Mathematica

CoefficientList[Series[1 / ((1 - 2 x) (1 - 7 x) (1 - 12 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 26 2013 *)

PARI

x='x+O('x^99); Vec(1/((1-2*x)*(1-7*x)*(1-12*x))) \\ Altug Alkan, Sep 21 2018

Formula

a(n) = 2*2^n/25 - 49*7^n/25 + 72*12^n/25. - R. J. Mathar, Jun 23 2013

From Vincenzo Librandi, Jun 26 2013: (Start)

a(n) = 21*a(n-1) - 122*a(n-2) + 168*a(n-3).

a(n) = 19*a(n-1) - 84*a(n-2) + 2^n. (End)

From Seiichi Manyama, May 04 2025: (Start)

a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).

a(n) = Sum_{k=0..n} (-5)^k * 12^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

E.g.f.: exp(2*x)*(2 - 49*exp(5*x) + 72*exp(10*x))/25. - Stefano Spezia, May 04 2025