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a(n) = (1/n!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_n=1..n} (-1)^(i_1 + i_2 + ... + i_n) * multinomial(i_1 + i_2 + ... + i_n; i_1, i_2, ..., i_n).
2

%I #17 May 22 2019 21:03:35

%S 1,-1,1,-120,1056496,-2063348839223,1457689055077930674637,

%T -569779896381467292745562607320194,

%U 174622933743914766946635359968704455433117668396,-56466564044341292662007179162722871704054012257606338926938133618

%N a(n) = (1/n!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_n=1..n} (-1)^(i_1 + i_2 + ... + i_n) * multinomial(i_1 + i_2 + ... + i_n; i_1, i_2, ..., i_n).

%H Seiichi Manyama, <a href="/A308327/b308327.txt">Table of n, a(n) for n = 0..27</a>

%e a(2) = (1/2) * (binomial(1+1,1) - binomial(1+2,2) - binomial(2+1,1) + binomial(2+2,2)) = 1.

%o (PARI) {a(n) = sum(i=n, n^2, (-1)^i*i!*polcoef(sum(j=1, n, x^j/j!)^n, i))/n!}

%Y Main diagonal of A308356.

%Y Cf. A308296.

%K sign

%O 0,4

%A _Seiichi Manyama_, May 20 2019