A348700 - OEIS

%I #20 Oct 31 2021 02:47:39

%S 1,24,1440,97440,6745200,467170704,32179283136,2201392866816,

%T 149582010088368,10100991172786800,678330750569025840,

%U 45330886561301259360,3016323760677017743680,199948320909528951802560,13210188418741950461761920,870202858863529042485373440

%N a(n) = Sum_{x_1+x_2+x_3+x_4=n, 0 <= x_1, x_2, x_3, x_4 <= n} (3*n)!/((n-x_1)! * (n-x_2)! * (n-x_3)! * (n-x_4)!).

%H Vaclav Kotesovec, <a href="/A348700/b348700.txt">Table of n, a(n) for n = 0..250</a>

%F From _Vaclav Kotesovec_, Oct 30 2021: (Start)

%F Recurrence: 2*(n-1)*n^2*(2*n - 1)*(65*n^5 - 1089*n^4 + 6575*n^3 - 18383*n^2 + 23840*n - 11584)*a(n) = (n-1)*(46475*n^8 - 823810*n^7 + 5577254*n^6 - 19173000*n^5 + 36602535*n^4 - 39648950*n^3 + 23762600*n^2 - 7257792*n + 875520)*a(n-1) - 6*(3*n - 5)*(3*n - 4)*(51155*n^7 - 830003*n^6 + 5035067*n^5 - 15152773*n^4 + 24453658*n^3 - 20908896*n^2 + 8592448*n - 1337856)*a(n-2) + 10368*(3*n - 8)*(3*n - 7)*(3*n - 5)*(3*n - 4)*(65*n^5 - 764*n^4 + 2869*n^3 - 4542*n^2 + 2768*n - 576)*a(n-3).

%F a(n) ~ 64^n. (End)

%o (PARI) a(n) = sum(a=0, n, sum(b=0, n, sum(c=0, n, sum(d=0, n, if(a+b+c+d==n, (3*n)!/((n-a)!*(n-b)!*(n-c)!*(n-d)!), 0)))));

%Y Cf. A000079, A092472, A348703.

%Y Cf. A348701.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 30 2021