A020726 Expansion of g.f. 1/((1-6*x)*(1-10*x)*(1-11*x)).

1, 27, 493, 7599, 106645, 1411431, 17955757, 222093423, 2690508229, 32080473975, 377794514461, 4405195463487, 50953884924853, 585473143132359, 6690087028209805, 76090252032830991, 861988540696279717, 9731848557669909783, 109550181794434004989, 1230051085699164039135

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (27,-236,660).

Crossrefs

Cf. A016173, A016174.

Programs

Mathematica

CoefficientList[Series[1/((1-6x)(1-10x)(1-11x)),{x,0,30}],x] (* or *) LinearRecurrence[{27,-236,660},{1,27,493},30] (* Harvey P. Dale, Oct 01 2014 *)

PARI

Vec(1/((1-6*x)*(1-10*x)*(1-11*x)) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

Formula

a(n) = 21*a(n-1) - 110*a(n-2) + 6^n for n>1, a(0)=1, a(1)=27. - Vincenzo Librandi, Mar 11 2011

a(n) = (4*11^(n+2) - 5*10^(n+2) + 6^(n+2))/20. - Yahia Kahloune, Jun 30 2013

In general, for the expansion of 1/((1-r*x)(1-s*x)(1-t*x)) with t > s > r, we have the formula: a(n) = ((s-r)*t^(n+2) - (t-r)*s^(n+2) + (t-s)*r^(n+2))/((s-r)*(t-r)*(t-s)). - Yahia Kahloune, Sep 09 2013

a(0) = 1, a(1) = 27, a(2) = 493, a(n) = 27*a(n-1) - 236*a(n-2) + 660*a(n-3). - Harvey P. Dale, Oct 01 2014

From Elmo R. Oliveira, Mar 26 2025: (Start)

E.g.f.: exp(6*x)*(484*exp(5*x) - 500*exp(4*x) + 36)/20.

a(n) = A016174(n+1) - A016173(n+1). (End)

Extensions

More terms from Elmo R. Oliveira, Mar 26 2025