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A111926 Expansion of x^4/((1-2*x)*(x^2-x+1)*(x-1)^2). 1

%I

%S 0,0,0,0,1,5,15,36,78,162,331,671,1353,2718,5448,10908,21829,43673,

%T 87363,174744,349506,699030,1398079,2796179,5592381,11184786,22369596,

%U 44739216,89478457,178956941,357913911,715827852,1431655734,2863311498

%N Expansion of x^4/((1-2*x)*(x^2-x+1)*(x-1)^2).

%C Binomial transform of sequence (0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0). Note: the binomial transform of the sequence (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A111927; the binomial transform of the sequence (0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A024495 (disregarding first two terms, which are both zero).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,11,-7,2).

%F a(n+2) - a(n+1) + a(n) = A000295(n) = 2^n - n - 1 (Eulerian numbers); a(n) = 1/3*2^n-n+2/3*(1/2+1/2*I*sqrt(3))^n*(-1/4-1/4*I*sqrt(3))+2/3*(1/2-1/2*I*sqrt(3))^n*(-1/4+1/4*I*sqrt(3))

%F a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(n)=5*a(n-1)-10*a(n-2)+ 11*a(n-3)- 7*a(n-4)+2*a(n-5). - _Harvey P. Dale_, Feb 24 2016

%t CoefficientList[Series[x^4/((1-2x)(x^2-x+1)(x-1)^2),{x,0,40}],x] (* or *) LinearRecurrence[{5,-10,11,-7,2},{0,0,0,0,1},40] (* _Harvey P. Dale_, Feb 24 2016 *)

%o Floretion Algebra Multiplication Program, FAMP Code: -4ibaseisumseq[ + .5'i + .5'j + .5'k + .5'ij' + .5'jk' + .5'ki' + e], sumtype: Y[8] = (int)Y[6] - (int)Y[7] + Y[8] + sum (internal program code).

%Y Cf. A000295, A111927, A024495.

%K easy,nonn

%O 0,6

%A _Creighton Dement_, Aug 21 2005

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Last modified February 26 20:23 EST 2020. Contains 332295 sequences. (Running on oeis4.)