A036219 Expansion of 1/(1-3*x)^6; 6-fold convolution of A000244 (powers of 3).

1, 18, 189, 1512, 10206, 61236, 336798, 1732104, 8444007, 39405366, 177324147, 773778096, 3288556908, 13660159464, 55616363532, 222465454128, 875957725629, 3400777052442, 13036312034361, 49400761393368, 185252855225130, 688082033693340, 2533392942234570

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (18,-135,540,-1215,1458,-729).

Crossrefs

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), this sequence (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

Programs

Magma

[3^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple

seq(3^n*binomial(n+5,5), n=0..30); # Zerinvary Lajos, Jun 13 2008

Mathematica

Table[3^n*Binomial[n+5, 5], {n, 0, 30}] (* G. C. Greubel, May 19 2021 *)
CoefficientList[Series[1/(1-3x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {18,-135,540,-1215,1458,-729},{1,18,189,1512,10206,61236},30] (* Harvey P. Dale, Jan 02 2022 *)

Sage

[3^n*binomial(n+5,5) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

Formula

a(n) = 3^n*binomial(n+5, 5).

a(n) = A027465(n+6, 6).

G.f.: 1/(1-3*x)^6.

E.g.f.: (1/40)*(40 + 600*x + 1800*x^2 + 1800*x^3 + 675*x^4 + 81*x^5)*exp(3*x). - G. C. Greubel, May 19 2021

From Amiram Eldar, Sep 22 2022: (Start)

Sum_{n>=0} 1/a(n) = 240*log(3/2) - 385/4.

Sum_{n>=0} (-1)^n/a(n) = 3840*log(4/3) - 4415/4. (End)