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a(n) = n! * [x^n] 1/(1 - 3*x)^(n/3).
5

%I #8 Aug 16 2018 06:02:26

%S 1,1,10,162,3640,104720,3674160,152152000,7264216960,392841187200,

%T 23734494784000,1584471003315200,115825295634048000,

%U 9201578813819392000,789383453851632640000,72728093032166347776000,7162140885524461957120000,750766815289210771251200000

%N a(n) = n! * [x^n] 1/(1 - 3*x)^(n/3).

%H G. C. Greubel, <a href="/A303486/b303486.txt">Table of n, a(n) for n = 0..343</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F a(n) = Product_{k=0..n-1} (3*k + n).

%F a(n) = 3^n*Gamma(4*n/3)/Gamma(n/3).

%F a(n) ~ 2^(8*n/3-1)*n^n/exp(n).

%e a(1) = 1;

%e a(2) = 2*5 = 10;

%e a(3) = 3*6*9 = 162;

%e a(4) = 4*7*10*13 = 3640;

%e a(5) = 5*8*11*14*17 = 104720, etc.

%t Table[n! SeriesCoefficient[1/(1 - 3 x)^(n/3), {x, 0, n}], {n, 0, 17}]

%t Table[Product[3 k + n, {k, 0, n - 1}], {n, 0, 17}]

%t Table[3^n Pochhammer[n/3, n], {n, 0, 17}]

%Y Column k=3 of A303489.

%Y Cf. A000407, A007559, A008544, A032031, A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609, A113551, A303487, A303488.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 24 2018