A016234 Expansion of 1/((1-x) * (1-5*x) * (1-9*x)).
1, 15, 166, 1650, 15631, 144585, 1320796, 11984820, 108351661, 977606355, 8810664226, 79357013190, 714518294491, 6432190529325, 57897344158456, 521114244398760, 4690218934452121, 42212924084385495, 379921085131051486, 3419313608037373530, 30773941681625912551
Offset: 0
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (15, -59, 45).
Programs
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Mathematica
CoefficientList[Series[1/((1-x)(1-5x)(1-9x)),{x,0,30}],x] (* or *) LinearRecurrence[{15,-59,45},{1,15,166},30] (* Harvey P. Dale, Oct 16 2014 *) -
PARI
Vec(1/((1-x)*(1-5*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
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PARI
a(n) = (9^(n+2) - 2*5^(n+2) + 1)/32; \\ Joerg Arndt, Aug 13 2013
Formula
a(0)=1, a(1)=15, a(n) = 14*a(n-1) - 45*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011
a(n) = (9^(n+2) - 2*5^(n+2) + 1)/32. - Yahia Kahloune, Aug 13 2013
a(0)=1, a(1)=15, a(2)=166, a(n) = 15*a(n-1) - 59*a(n-2) + 45*a(n-3). - Harvey P. Dale, Oct 16 2014
O.g.f.: see the name.
E.g.f.: (d^2/dx^2) (exp(x)*((exp(4*x) - 1)^2)/(4^2*2!)) = exp(x)*(1 - 50*exp(4*x) + 81*exp(8*x))/32.
From Seiichi Manyama, May 05 2025: (Start)
a(n) = Sum_{k=0..n} 4^k * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-4)^k * 9^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)