A036223 Expansion of 1/(1-3*x)^10; 10-fold convolution of A000244 (powers of 3).

1, 30, 495, 5940, 57915, 486486, 3648645, 25019280, 159497910, 956987460, 5454828522, 29753610120, 156206453130, 793048146660, 3908594437110, 18761253298128, 87943374834975, 403504896301650, 1815772033357425

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

With a different offset, number of n-permutations (n >= 9) of 4 objects: u, v, z, x with repetition allowed, containing exactly nine (9) u's. - Zerinvary Lajos, Jul 02 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049).

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), A036221 (m=7), A036222 (m=8), this sequence (m=9), A172362 (m=10).

Programs

Magma

[3^n*Binomial(n+9, 9): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple

seq(3^n*binomial(n+9, 9), n=0..20); # Zerinvary Lajos, Jul 02 2008

Mathematica

Table[3^n*Binomial[n+9,9], {n,0,30}] (* G. C. Greubel, May 18 2021 *)
CoefficientList[Series[1/(1-3x)^10,{x,0,30}],x] (* or *) LinearRecurrence[ {30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049},{1,30,495,5940,57915,486486,3648645,25019280,159497910,956987460},30] (* Harvey P. Dale, Jan 16 2022 *)

Sage

[3^n*binomial(n+9,9) for n in range(30)] # Zerinvary Lajos, Mar 13 2009

Formula

a(n) = 3^n*binomial(n+9, 9).

a(n) = A027465(n+10, 10).

G.f.: 1/(1-3*x)^10.

E.g.f.: (4480 + 120960*x + 725760*x^2 + 1693440*x^3 + 1905120*x^4 + 1143072*x^5 + 381024*x^6 + 69984*x^7 + 6561*x^8 + 243*x^9)*exp(3*x)/4480. - G. C. Greubel, May 18 2021

From Amiram Eldar, Sep 22 2022: (Start)

Sum_{n>=0} 1/a(n) = 6912*log(3/2) - 784431/280.

Sum_{n>=0} (-1)^n/a(n) = 1769472*log(4/3) - 142532433/280. (End)