A178759 Expansion of e.g.f. 3*x*exp(x)*(exp(x)-1)^2.

0, 0, 0, 18, 144, 750, 3240, 12642, 46368, 163350, 559800, 1881066, 6229872, 20406750, 66273480, 213759090, 685601856, 2188698150, 6959413080, 22053083514, 69672773520, 219535296750, 690106487400, 2164714299138, 6777100916064, 21179698653750, 66083277045240, 205880260458762

Offset

0,4

Comments

a(n) is the sum of the digits in ternary sequences of length n, in which each element of the alphabet, {0,1,2} appears at least once in the sequence.

Generally, the e.g.f. for such sum of n-ary sequences (taken on an alphabet of {0,1,2,...,n-1}) is binomial(n,2)*x*exp(x)*(exp(x)-1)^(n-1).

Cf. A058877 which is the sum of the digits in such binary sequences.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (12,-58,144,-193,132,-36).

Formula

E.g.f.: 3*x*exp(x)*(exp(x)-1)^2.

a(n) = (3^n - 3*2^n + 3)*n. - Mark van Hoeij, May 13 2013

G.f.: 6*x^3*(11*x^2-12*x+3) / ((x-1)^2*(2*x-1)^2*(3*x-1)^2). - Colin Barker, Nov 30 2014

Example

a(3)=18 because there are six length 3 sequences on {0,1,2} that contain at least one 0, at least one 1 and at least one 2: (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1),(2,1,0).  The digits sum to 18.

Mathematica

Range[0, 20]! CoefficientList[Series[3x Exp[x](Exp[x]-1)^2, {x, 0, 20}], x]
Table[(3^n -3*2^n +3)*n, {n, 0, 30)] (* G. C. Greubel, Jan 24 2019 *)

Prog

(PARI) x='x+O('x^66); concat([0, 0, 0], Vec(serlaplace(3*x*exp(x)*(exp(x)-1)^2))) \\ Joerg Arndt, May 13 2013
(PARI) concat([0, 0, 0], Vec(6*x^3*(11*x^2-12*x+3)/((x-1)^2*(2*x-1)^2*(3*x-1)^2) + O(x^100))) \\ Colin Barker, Nov 30 2014
(PARI) vector(30, n, n--; (3^n - 3*2^n + 3)*n) \\ G. C. Greubel, Jan 24 2019
(Magma) [(3^n - 3*2^n + 3)*n: n in [0..30]]; // G. C. Greubel, Jan 24 2019
(Sage) [(3^n - 3*2^n + 3)*n for n in (0..30)] # G. C. Greubel, Jan 24 2019
(GAP) List([0..30], n -> (3^n - 3*2^n + 3)*n); # G. C. Greubel, Jan 24 2019

Crossrefs

Cf. A058877, A178756.

Sequence in context: A127408 A008452 A126900 * A036397 A247741 A224329

Adjacent sequences: A178756 A178757 A178758 * A178760 A178761 A178762

Keyword

nonn,easy

Author

Geoffrey Critzer, Dec 26 2010

Status

approved