A028037 - OEIS

%I #38 Jul 25 2024 23:03:35

%S 1,25,415,5845,76111,952525,11679655,141710965,1710104671,20579684125,

%T 247310795095,2969873300485,35651407676431,427894724193325,

%U 5135204742169735,61625269469056405,739520126057523391

%N Expansion of 1/((1-3x)(1-4x)(1-6x)(1-12x)).

%H Aung Phone Maw and Aung Kyaw, <a href="https://arxiv.org/abs/1711.10716">Recursive Harmonic Numbers and Binomial Coefficients</a>, arXiv:1711.10716 [math.CO], 2017.

%H Jerry Metzger and Thomas Richards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Metzger/metz1.html">A Prisoner Problem Variation</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (25,-210,720,-864).

%F a(n) = (12^(n+3) - 12*6^(n+3) + 27*4^(n+3) - 16*3^(n+3))/432 = -3^n + 4^(n+1) + 3^n*4^(n+1) - 6^(n+1). - _Yahia Kahloune_, May 31 2013

%F a(n) = 25*a(n-1) - 210*a(n-2) + 720*a(n-3) - 864*a(n-4), with a(0)=1, a(1)=25, a(2)=415, a(3)=5845. - _Harvey P. Dale_, May 23 2014

%F G.f.: 12*x/(5*binomial(12*x,5)). - _Vladimir Kruchinin_, Apr 17 2016

%F E.g.f.: (-1 + 4*exp(x) - 6*exp(3*x) + 4*exp(9*x))*exp(3*x). - _Ilya Gutkovskiy_, Apr 17 2016

%t CoefficientList[Series[1/((1-3x)(1-4x)(1-6x)(1-12x)),{x,0,30}],x] (* or *) LinearRecurrence[{25,-210,720,-864},{1,25,415,5845},30] (* _Harvey P. Dale_, May 23 2014 *)

%o (PARI) Vec(1/((1-3*x)*(1-4*x)*(1-6*x)*(1-12*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012

%o (Maxima)

%o taylor(12*x/binomial(12*x,5)/5,x,0,10); /* _Vladimir Kruchinin_, Apr 17 2016 */

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_