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

%I A111926 #19 Jan 03 2025 05:10:17
%S A111926 0,0,0,0,1,5,15,36,78,162,331,671,1353,2718,5448,10908,21829,43673,
%T A111926 87363,174744,349506,699030,1398079,2796179,5592381,11184786,22369596,
%U A111926 44739216,89478457,178956941,357913911,715827852,1431655734,2863311498
%N A111926 Expansion of x^4/((1-2*x)*(x^2-x+1)*(x-1)^2).
%C A111926 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).
%C A111926 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).
%H A111926 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,11,-7,2).
%F A111926 a(n+2) - a(n+1) + a(n) = A000295(n) = 2^n - n - 1 (Eulerian numbers).
%F A111926 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 A111926 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
%F A111926 a(n) = Sum_{k=1..floor(n/2)} binomial(n, 3*k+1). - _Taras Goy_, Jan 02 2025
%F A111926 E.g.f.: exp(x/2)*(exp(x/2)*(exp(x) - 3*x) - cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - _Stefano Spezia_, Jan 03 2025
%t A111926 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 *)
%Y A111926 Cf. A000295, A111927, A024495.
%K A111926 easy,nonn
%O A111926 0,6
%A A111926 _Creighton Dement_, Aug 21 2005