A016209 Expansion of 1/((1-x)(1-3x)(1-5x)).

1, 9, 58, 330, 1771, 9219, 47188, 239220, 1205941, 6059229, 30384718, 152189310, 761743711, 3811110039, 19062724648, 95335146600, 476740303081, 2383895225649, 11920057258978, 59602029687090

Offset

0,2

Comments

For a combinatorial interpretation following from a(n) = A039755(n+2,2) = h^{(3)}_n, the complete homogeneous symmetric function of degree n in the symbols {1, 3, 5} see A039755. - Wolfdieter Lang, May 26 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Formula

a(n) = A039755(n+2, 2).

a(n) = (5^(n+2) - 2*3^(n+2)+1)/8 = a(n-1) + A005059(n+1) = 8*a(n-1) - 15*a(n-2) + 1 = (A003463(n+2) - A003462(n+2))/2. - Henry Bottomley, Jun 06 2000

G.f.: 1/((1-x)(1-3*x)(1-5*x)). See the name.

E.g.f.: (25*exp(5*x) - 18*exp(3*x) + exp(x))/8, from the e.g.f. of the third column (k=2) of A039755. - Wolfdieter Lang, May 26 2017

Example

a(2) = h^{(3)}_2 = 1^2 + 3^2 + 5^2 + 1^1*(3^1 + 5^1) + 3^1*5^1 = 58. - Wolfdieter Lang, May 26 2017

Maple

A016209 := proc(n) (5^(n+2)-2*3^(n+2)+1)/8; end proc: # R. J. Mathar, Mar 22 2011

Mathematica

Join[{a=1, b=9}, Table[c=8*b-15*a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2011 *)
CoefficientList[Series[1/((1-x)(1-3x)(1-5x)), {x, 0, 30}], x] (* or *) LinearRecurrence[ {9, -23, 15}, {1, 9, 58}, 30] (* Harvey P. Dale, Feb 20 2020 *)

Prog

(PARI) a(n)=if(n<0, 0, n+=2; (5^n-2*3^n+1)/8)
(Magma) [(5^(n+2)-2*3^(n+2)+1)/8: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

Crossrefs

Cf. A016218, A016208, A000392, A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A016256, A039755, A021424.

Sequence in context: A026750 A009034 A026377 * A196920 A129173 A271271

Adjacent sequences: A016206 A016207 A016208 * A016210 A016211 A016212

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved