A016175 Expansion of 1/((1-6*x)*(1-12*x)).
1, 18, 252, 3240, 40176, 489888, 5925312, 71383680, 858283776, 10309483008, 123774262272, 1485653944320, 17830024114176, 213973350064128, 2567758564933632, 30813572964188160, 369765696680165376, 4437205286821429248, 53246565001813819392, 638959389381505843200, 7667516328736510181376, 92010217881788762554368
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..920
- Index entries for linear recurrences with constant coefficients, signature (18,-72).
Programs
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Magma
[2*12^n - 6^n: n in [0..40]]; // G. C. Greubel, Nov 13 2024
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Mathematica
Table[2*12^n -6^n, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *) LinearRecurrence[{18,-72},{1,18},40] (* Harvey P. Dale, Nov 25 2013 *) -
PARI
Vec(1/((1-6*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
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SageMath
A016175= BinaryRecurrenceSequence(18,-72,1,18) print([A016175(n) for n in range(41)]) # G. C. Greubel, Nov 13 2024
Formula
a(n) = (6^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = 2*12^n - 6^n.
E.g.f.: (d^2/dx^2)((((exp(6*x)-1)/6)^2)/2!) = -exp(6*x) + 2*exp(12*x).
a(n) = 3^n*binomial(2^(n+1), 2). - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
a(n) = 12*a(n-1) + 6^n, n >= 1. - Vincenzo Librandi, Feb 09 2011
a(n) = 18*a(n-1) - 72*a(n-2), n >= 2. - Vincenzo Librandi, Feb 09 2011
Extensions
More terms added by G. C. Greubel, Nov 13 2024