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

0,2

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).

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)

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 *)

Prog

(Sage) [3^n*binomial(n+4, 4) for n in range(30)] # Zerinvary Lajos, Mar 10 2009
(Magma) [3^n* Binomial(n+4, 4): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

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).

Sequence in context: A027630 A027629 A023013 * A022643 A125378 A254409

Adjacent sequences: A036214 A036215 A036216 * A036218 A036219 A036220

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved