A028037 Expansion of 1/((1-3x)(1-4x)(1-6x)(1-12x)).

1, 25, 415, 5845, 76111, 952525, 11679655, 141710965, 1710104671, 20579684125, 247310795095, 2969873300485, 35651407676431, 427894724193325, 5135204742169735, 61625269469056405, 739520126057523391

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Aung Phone Maw and Aung Kyaw, Recursive Harmonic Numbers and Binomial Coefficients, arXiv:1711.10716 [math.CO], 2017.
• Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
• Index entries for linear recurrences with constant coefficients, signature (25,-210,720,-864).

Programs

Mathematica

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 *)

Maxima

taylor(12*x/binomial(12*x,5)/5,x,0,10); /* Vladimir Kruchinin, Apr 17 2016 */

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

Formula

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

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

G.f.: 12*x/(5*binomial(12*x,5)). - Vladimir Kruchinin, Apr 17 2016

E.g.f.: (-1 + 4*exp(x) - 6*exp(3*x) + 4*exp(9*x))*exp(3*x). - Ilya Gutkovskiy, Apr 17 2016