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 A242095 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns or the symbol set; square array A(n,k), n>=0, k>=0, read by antidiagonals. 15
 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 5, 1, 0, 1, 1, 8, 18, 1, 0, 1, 1, 9, 139, 173, 1, 0, 1, 1, 9, 408, 15412, 2812, 1, 0, 1, 1, 9, 649, 332034, 10805764, 126446, 1, 0, 1, 1, 9, 749, 2283123, 3327329224, 50459685390, 16821330, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS A(n,k) = A(n,k+1) for k >= n^2. LINKS Alois P. Heinz, Antidiagonals n = 0..20, flattened EXAMPLE A(2,2) = 5: [1 1] [2 1] [2 2] [2 1] [2 1] [1 1], [1 1], [1 1], [2 1], [1 2]. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 0, 1, 1, 1, 1, 1, ... 0, 1, 5, 8, 9, 9, ... 0, 1, 18, 139, 408, 649, ... 0, 1, 173, 15412, 332034, 2283123, ... 0, 1, 2812, 10805764, 3327329224, 173636442196, ... MAPLE with(numtheory): b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {}, {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)})) end: A:= proc(n, k) option remember; add(add(add(mul(mul(add(d* coeff(u, x, d), d=divisors(ilcm(i, j)))^(igcd(i, j)* coeff(s, x, i)*coeff(t, x, j)), j=1..degree(t)), i=1..degree(s))/mul(i^coeff(u, x, i)*coeff(u, x, i)!, i=1..degree(u))/mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t))/mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s)), u=b(k\$2)), t=b(n\$2)), s=b(n\$2)) end: seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA 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}]]]; A[n_, k_] := A[n, k] = Sum[Sum[Sum[Product[Product[With[{g = GCD[i, j]* Coefficient[s, x, i]*Coefficient[t, x, j]}, If[g == 0, 1, Sum[d* Coefficient[u, x, d], {d, Divisors[LCM[i, j]]}]^g]], {j, Exponent[t, x]} ], {i, Exponent[s, x]}]/Product[i^Coefficient[u, x, i]*Coefficient[u, x, i]!, {i, Exponent[u, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, Exponent[t, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, Exponent[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz, updated Jan 01 2021 *) CROSSREFS Columns k=0-10 give: A000007, A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126. Main diagonal gives A091058. A(n,n^2) gives A091057. Cf. A242093, A242106, A246106. Sequence in context: A281563 A293087 A209349 * A336169 A007912 A019755 Adjacent sequences: A242092 A242093 A242094 * A242096 A242097 A242098 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 14 2014 STATUS approved

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Last modified December 6 01:03 EST 2022. Contains 358594 sequences. (Running on oeis4.)