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

0, 0, 0, 0, 1, 5, 15, 36, 78, 162, 331, 671, 1353, 2718, 5448, 10908, 21829, 43673, 87363, 174744, 349506, 699030, 1398079, 2796179, 5592381, 11184786, 22369596, 44739216, 89478457, 178956941, 357913911, 715827852, 1431655734, 2863311498

Offset: 0

Author

Creighton Dement, Aug 21 2005

Keywords

• easy
• nonn

Comments

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).

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).

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-7,2).

Crossrefs

Cf. A000295, A111927, A024495.

Programs

Mathematica

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 *)

Formula

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)).

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

a(n) = Sum_{k=1..floor(n/2)} binomial(n, 3*k+1). - Taras Goy, Jan 02 2025

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