A036216 Expansion of 1/(1 - 3*x)^4; 4-fold convolution of A000244 (powers of 3).

1, 12, 90, 540, 2835, 13608, 61236, 262440, 1082565, 4330260, 16888014, 64481508, 241805655, 892820880, 3252418920, 11708708112, 41712272649, 147219785820, 515269250370, 1789882659180, 6175095174171, 21171754882872

Offset

0,2

Comments

With three leading zeros, 3rd binomial transform of (0,0,0,1,0,0,0,0,...). - Paul Barry, Mar 07 2003

Number of n-permutations (n=4) of 4 objects u, v, w, z, with repetition allowed, containing exactly three u's. - Zerinvary Lajos, May 23 2008

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Eric Weisstein's World of Mathematics, Idempotent Number.

Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Formula

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

a(n) = A027465(n+4, 4).

G.f.: 1/(1 - 3*x)^4.

With three leading zeros, a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4), a(0) = a(1) = a(2) = 0, a(3) = 1. - Paul Barry, Mar 07 2003

With three leading zeros, C(n, 3)*3^(n-3) is the second binomial transform of C(n, 3). - Paul Barry, Jul 24 2003

E.g.f.: (1/2)*(2 + 18*x + 27*x^2 + 9*x^3)*exp(3*x). - Franck Maminirina Ramaharo, Nov 23 2018

From Amiram Eldar, Jan 05 2022: (Start)

Sum_{n>=0} 1/a(n) = 36*log(3/2) - 27/2.

Sum_{n>=0} (-1)^n/a(n) = 144*log(4/3) - 81/2. (End)

Maple

seq(3^n*binomial(n+3, 3), n=0..30)]; # Zerinvary Lajos, Dec 21 2006

Mathematica

CoefficientList[Series[1/(1-3x)^4, {x, 0, 30}], x] (* or *) LinearRecurrence[ {12, -54, 108, -81}, {1, 12, 90, 540}, 30] (* Harvey P. Dale, Jul 27 2017 *)

Prog

(Sage) [3^n*binomial(n+3, 3) for n in range(30)] # Zerinvary Lajos, Mar 10 2009
(Magma) [3^n* Binomial(n+3, 3): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011
(PARI) a(n) = 3^n*binomial(n+3, 3) \\ Charles R Greathouse IV, Oct 03 2016

Crossrefs

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), this sequence (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

Sequence in context: A130072 A135158 A073382 * A022640 A090749 A130592

Adjacent sequences: A036213 A036214 A036215 * A036217 A036218 A036219

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved