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A336172 a(n) = Sum_{k=0..n} (-1)^(n-k) * multinomial(n+(n-1)*k; n-k, {k}^n).

%I A336172 #19 Jul 11 2020 02:33:52
%S A336172 1,0,1,1109,58370761,616967236620839,2667424979594724623421841,
%T A336172 7363013376580445620328429354795257679,
%U A336172 18165626590057176919253932602358956717053291818375681,53130672834302729155813211757017136527240145191602729876485547920535679
%N A336172 a(n) = Sum_{k=0..n} (-1)^(n-k) * multinomial(n+(n-1)*k; n-k, {k}^n).
%H A336172 Seiichi Manyama, <a href="/A336172/b336172.txt">Table of n, a(n) for n = 0..26</a>
%F A336172 a(n) = [x^n] Sum_{k=0..n} (n*k)!/k!^n * x^k / (1+x)^(n*k+1).
%t A336172 a[n_] := Sum[(-1)^(n - k)*(n + (n - 1)*k)!/((n - k)!*k!^n), {k, 0, n}]; Array[a, 10, 0] (* _Amiram Eldar_, Jul 10 2020 *)
%o A336172 (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*(n+(n-1)*k)!/((n-k)!*k!^n))}
%Y A336172 Main diagonal of A336169.
%Y A336172 Cf. A229267.
%K A336172 nonn
%O A336172 0,4
%A A336172 _Seiichi Manyama_, Jul 10 2020