A036217 Expansion of 1/(1-3*x)^5; 5-fold convolution of A000244 (powers of 3).

1, 15, 135, 945, 5670, 30618, 153090, 721710, 3247695, 14073345, 59108049, 241805655, 967222620, 3794488740, 14635885140, 55616363532, 208561363245, 772903875555, 2833980877035, 10291825290285, 37050571045026

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

With a different offset, number of n-permutations (n=5) of 4 objects: u, v, z, x with repetition allowed, containing exactly four (4) u's. Example: a(1)=15 because we have uuuuv uuuvu uuvuu uvuuu vuuuu uuuuz uuuzu uuzuu uzuuu zuuuu uuuux uuuxu uuxuu uxuuu xuuuu. - Zerinvary Lajos, Jun 12 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (15,-90,270,-405,243).

Crossrefs

Cf. A000332, A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), this sequence (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

Programs

Magma

[3^n* Binomial(n+4, 4): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

Maple

seq(3^n*binomial(n+4,4), n=0..30); # Zerinvary Lajos, Jun 12 2008

Mathematica

CoefficientList[Series[1/(1-3x)^5,{x,0,30}],x] (* Harvey P. Dale, Jun 13 2017 *)

Sage

[3^n*binomial(n+4,4) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

Formula

a(n) = 3^n*binomial(n+4, 4) = 3^n*A000332(n+4).

a(n) = A027465(n+5, 5).

G.f.: 1/(1-3*x)^5.

E.g.f.: (1/8)*(8 +96*x +216*x^2 +144*x^3 +27*x^4)*exp(3*x). - G. C. Greubel, May 19 2021

From Amiram Eldar, Sep 22 2022: (Start)

Sum_{n>=0} 1/a(n) = 40 - 96*log(3/2).

Sum_{n>=0} (-1)^n/a(n) = 768*log(4/3) - 220. (End)