A036221 Expansion of 1/(1-3*x)^8; 8-fold convolution of A000244 (powers of 3).

1, 24, 324, 3240, 26730, 192456, 1250964, 7505784, 42220035, 225173520, 1148384952, 5637526128, 26778249108, 123591918960, 556163635320, 2447119995408, 10553204980197, 44695926974952, 186233029062300

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

With a different offset, number of n-permutations (n>=7) of 4 objects: u, v, z, x with repetition allowed, containing exactly seven (7) u's. Example: a(1)=24 because we have uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu. - Zerinvary Lajos, Jun 23 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (24,-252,1512,-5670,13608,-20412,17496,-6561).

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), A036219 (m=5), A036220 (m=6), this sequence (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

Programs

Magma

[3^n*Binomial(n+7, 7): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple

seq(3^n*binomial(n+7,7), n=0..30); # Zerinvary Lajos, Jun 23 2008

Mathematica

Table[3^n*Binomial[n+7,7], {n,0,30}] (* G. C. Greubel, May 19 2021 *)

Sage

[3^n*binomial(n+7, 7) for n in range(30)] # Zerinvary Lajos, Mar 13 2009

Formula

a(n) = 3^n*binomial(n+7, 7).

a(n) = A027465(n+8, 8.)

G.f.: 1/(1-3*x)^8.

E.g.f.: (1/560)*(560 +11760*x +52920*x^2 +88200*x^3 +66150*x^4 +23814*x^5 +3969*x^6 +243*x^7)*exp(3*x). - G. C. Greubel, May 19 2021

From Amiram Eldar, Sep 22 2022: (Start)

Sum_{n>=0} 1/a(n) = 1344*log(3/2) - 5439/10.

Sum_{n>=0} (-1)^n/a(n) = 86016*log(4/3) - 247443/10. (End)