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Expansion of 1/(1-3*x)^10; 10-fold convolution of A000244 (powers of 3).
12

%I #30 Sep 22 2022 01:56:01

%S 1,30,495,5940,57915,486486,3648645,25019280,159497910,956987460,

%T 5454828522,29753610120,156206453130,793048146660,3908594437110,

%U 18761253298128,87943374834975,403504896301650,1815772033357425

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

%C 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

%H Vincenzo Librandi, <a href="/A036223/b036223.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049).

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

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

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

%F 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

%F From _Amiram Eldar_, Sep 22 2022: (Start)

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

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

%p seq(3^n*binomial(n+9, 9), n=0..20); # _Zerinvary Lajos_, Jul 02 2008

%t Table[3^n*Binomial[n+9,9], {n,0,30}] (* _G. C. Greubel_, May 18 2021 *)

%t 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 *)

%o (Sage) [3^n*binomial(n+9,9) for n in range(30)] # _Zerinvary Lajos_, Mar 13 2009

%o (Magma) [3^n*Binomial(n+9, 9): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011

%Y Cf. A027465.

%Y 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).

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_