A016307 Expansion of g.f. 1/((1-2*x)*(1-6*x)*(1-10*x)).
1, 18, 232, 2640, 28336, 295008, 3020032, 30620160, 308720896, 3102325248, 31113951232, 311683706880, 3120102240256, 31220613439488, 312323680632832, 3123942083788800, 31243652502716416, 312461915016265728, 3124771490097528832, 31248628940585041920
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (18,-92,120).
Programs
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Magma
[(2^n-18*6^n+25*10^n)/8: n in [0..20]]; // Vincenzo Librandi, Sep 01 2011
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Mathematica
CoefficientList[Series[1/((1-2x)(1-6x)(1-10x)),{x,0,30}],x] (* or *) LinearRecurrence[{18,-92,120},{1,18,232},30] (* Harvey P. Dale, Nov 06 2019 *)
Formula
From Vincenzo Librandi, Sep 01 2011: (Start)
a(n) = (2^n - 18*6^n + 25*10^n)/8.
a(n) = 18*a(n-1) - 92*a(n-2) + 120*a(n-3) for n > 2.
a(n) = 16*a(n-1) - 60*a(n-2) + 2^n for n > 1. (End)
From Seiichi Manyama, May 04 2025: (Start)
a(n) = Sum_{k=0..n} 4^k * 2^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-4)^k * 10^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)
E.g.f.: exp(2*x)*(1 - 18*exp(4*x) + 25*exp(8*x))/8. - Stefano Spezia, May 04 2025