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A318951
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Array read by rows: T(n,k) is the number of nonisomorphic n X n matrices with nonnegative integer entries and row sums k under row and column permutations, (n >= 1, k >= 0).
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11
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1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 14, 5, 1, 1, 9, 44, 53, 7, 1, 1, 12, 129, 458, 198, 11, 1, 1, 16, 316, 3411, 5929, 782, 15, 1, 1, 20, 714, 19865, 145168, 96073, 3111, 22, 1, 1, 25, 1452, 95214, 2459994, 9283247, 1863594, 12789, 30, 1, 1, 30, 2775, 383714, 30170387, 537001197, 833593500, 42430061, 53836, 42, 1
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OFFSET
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1,5
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LINKS
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EXAMPLE
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Array begins:
================================================================
n\k| 0 1 2 3 4 5 6
---|------------------------------------------------------------
1 | 1 1 1 1 1 1 1 ...
2 | 1 2 4 6 9 12 16 ...
3 | 1 3 14 44 129 316 714 ...
4 | 1 5 53 458 3411 19865 95214 ...
5 | 1 7 198 5929 145168 2459994 30170387 ...
6 | 1 11 782 96073 9283247 537001197 19578605324 ...
7 | 1 15 3111 1863594 833593500 189076534322 23361610029905 ...
...
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MATHEMATICA
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permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
Table[RowSumMats[n-k, n-k, k], {n, 1, 11}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Sep 12 2018, after Andrew Howroyd *)
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PROG
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(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={polcoeff(1/prod(j=1, #q, my(g=gcd(t, q[j])); (1 - x^(q[j]/g) + O(x*x^k))^g), k)}
RowSumMats(n, m, k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoeff(exp(sum(t=1, n, K(q, t, k)/t*x^t) + O(x*x^n)), n)); s/m!}
for(n=1, 8, for(k=0, 6, print1(RowSumMats(n, n, k), ", ")); print)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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