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a(n) = Sum_{k=0..n} (-1)^k * binomial(n^2 - k,n*k).
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%I #15 Oct 13 2021 10:25:28

%S 1,1,-2,-48,1626,931040,-479909170,-5499596761127,43158334880135692,

%T 9081843551946977373216,-1095541637114516172591381711,

%U -4049135740387789992460066844854898,7569951149407063102291625516677078697579

%N a(n) = Sum_{k=0..n} (-1)^k * binomial(n^2 - k,n*k).

%H Seiichi Manyama, <a href="/A348322/b348322.txt">Table of n, a(n) for n = 0..57</a>

%F a(n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n + x^(n+1)).

%t a[n_] := Sum[(-1)^k * Binomial[n^2 - k, n*k], {k, 0, n}]; Array[a, 13, 0] (* _Amiram Eldar_, Oct 12 2021 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n^2-k, n*k));

%o (PARI) a(n) = polcoef((1-x)^(n-1)/((1-x)^n+x^(n+1)+x*O(x^n^2)), n^2);

%Y Cf. A078001, A348315, A348321.

%K sign

%O 0,3

%A _Seiichi Manyama_, Oct 12 2021