A349480 - OEIS

%I #16 Nov 20 2021 07:20:37

%S 1,1,10,390,33456,4845360,1059099840,325460948400,133697543616000,

%T 70733019878196480,46831083260349024000,37927830201482962540800,

%U 36883442511877368877747200,42409212946187708288828160000

%N a(n) = Sum_{j=0..n} (-1)^(n-j) * Product_{k=(j-1)*n+1..j*n} k.

%H Seiichi Manyama, <a href="/A349480/b349480.txt">Table of n, a(n) for n = 0..214</a>

%F a(n) = n! * A349470(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,n).

%e a(2) = -1*2 + 3*4 = 10.

%e a(3) = 1*2*3 - 4*5*6 + 7*8*9 = 390.

%e a(4) = -1*2*3*4 + 5*6*7*8 - 9*10*11*12 + 13*14*15*16 = 33456.

%t a[n_] := n! * Sum[(-1)^(n - k) * Binomial[k*n, n], {k, 0, n}]; Array[a, 14, 0] (* _Amiram Eldar_, Nov 19 2021 *)

%o (PARI) a(n) = sum(j=0, n, (-1)^(n-j)*prod(k=(j-1)*n+1, j*n, k));

%Y Cf. A336513, A349470.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 19 2021