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A091057
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Number of n X n matrices over symbol set {1,...,n^2} equivalent under any permutation of row, columns or the symbol set.
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9
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1, 1, 9, 777, 18500104, 322286625959257, 7368376339801908226685191, 422262377369187686156418513093399998333, 105882936532098986759153041871810253870024776751177723954
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n, 1*u_1+2*u_2+...=n^2} (fixA[s_1, s_2, ...; t_1, t_2, ...; u_1, u_2, ...]/ (1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...*1^u_1*u_1!*2^u_2*u_2!*...)) where fixA[...] = Product_{i,j>=1} ((Sum_{d|lcm(i, j)} (d*u_d))^(gcd(i, j)*s_i*t_j)). - corrected by Max Alekseyev, Jun 03 2023
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[Function[p, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
A242095[n_, k_] := A242095[n, k] = With[{co = Coefficient, ex = Exponent}, Sum[Sum[Sum[Product[Product[With[{g = GCD[i, j]*co[s, x, i]*co[t, x, j]}, If[g == 0, 1, Sum[d*co[u, x, d], {d, Divisors[LCM[i, j]]}]^g]], {j, ex[t, x]}], {i, ex[s, x]}]/Product[i^co[u, x, i]*co[u, x, i]!, {i, ex[u, x]}]/Product[i^co[t, x, i]*co[t, x, i]!, {i, ex[t, x]}]/Product[i^co[s, x, i]*co[s, x, i]!, {i, ex[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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