A016175 - OEIS

%I #28 Nov 14 2024 05:55:51

%S 1,18,252,3240,40176,489888,5925312,71383680,858283776,10309483008,

%T 123774262272,1485653944320,17830024114176,213973350064128,

%U 2567758564933632,30813572964188160,369765696680165376,4437205286821429248,53246565001813819392,638959389381505843200,7667516328736510181376,92010217881788762554368

%N Expansion of 1/((1-6*x)*(1-12*x)).

%H G. C. Greubel, <a href="/A016175/b016175.txt">Table of n, a(n) for n = 0..920</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-72).

%F a(n) = (6^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).

%F a(n) = 2*12^n - 6^n.

%F E.g.f.: (d^2/dx^2)((((exp(6*x)-1)/6)^2)/2!) = -exp(6*x) + 2*exp(12*x).

%F a(n) = 3^n*binomial(2^(n+1), 2). - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009

%F a(n) = 12*a(n-1) + 6^n, n >= 1. - _Vincenzo Librandi_, Feb 09 2011

%F a(n) = 18*a(n-1) - 72*a(n-2), n >= 2. - _Vincenzo Librandi_, Feb 09 2011

%t Table[2*12^n -6^n, {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 09 2011 *)

%t LinearRecurrence[{18,-72},{1,18},40] (* _Harvey P. Dale_, Nov 25 2013 *)

%o (PARI) Vec(1/((1-6*x)*(1-12*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Magma) [2*12^n - 6^n: n in [0..40]]; // _G. C. Greubel_, Nov 13 2024

%o (SageMath)

%o A016175= BinaryRecurrenceSequence(18,-72,1,18)

%o print([A016175(n) for n in range(41)]) # _G. C. Greubel_, Nov 13 2024

%Y Second column of triangle A075501.

%Y Cf. A016129, A075916.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms added by _G. C. Greubel_, Nov 13 2024