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

0,6

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

Table of n, a(n) for n=0..33.

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

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

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 *)

Crossrefs

Cf. A000295, A111927, A024495.

Sequence in context: A144898 A163250 A053808 * A369804 A137609 A109818

Adjacent sequences: A111923 A111924 A111925 * A111927 A111928 A111929

Keyword

easy,nonn

Author

Creighton Dement, Aug 21 2005

Status

approved