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

Author

Wolfdieter Lang

Keywords

• easy
• nonn

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).

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).

Programs

Magma

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

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 *)

PARI

a(n) = 3^n*binomial(n+3, 3) \\ Charles R Greathouse IV, Oct 03 2016

Sage

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

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)