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A265706
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Rectangular array A read by upward antidiagonals: A(n,m) is the number of total difunctional (regular) binary relations between an n-element set and an m-element set.
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2
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1, 3, 1, 7, 5, 1, 15, 19, 9, 1, 31, 65, 49, 17, 1, 63, 211, 225, 127, 33, 1, 127, 665, 961, 749, 337, 65, 1, 255, 2059, 3969, 3991, 2505, 919, 129, 1, 511, 6305, 16129, 20237, 16201, 8525, 2569, 257, 1, 1023, 19171, 65025, 100087, 97713, 65911
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OFFSET
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1,2
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COMMENTS
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A(n,m) is the number of total difunctional (regular) binary relations between an n-element set and an m-element set.
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LINKS
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FORMULA
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T(n, m) = Sum_{i=1..n} (Stirling2(m, i)* i! + Stirling2(m, i+1)*(i+1)!) *Stirling2(n, i).
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EXAMPLE
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Array A begins
1 3 7 15 31 63 127 255 511
1 5 19 65 211 665 2059 6305 19171
1 9 49 225 961 3969 16129 65025 261121
1 17 127 749 3991 20237 100087 489149 2379511
1 33 337 2505 16201 97713 568177 3242265 18341401
1 65 919 8525 65911 464645 3115519 20322605 130656871
1 129 2569 29625 271561 2214009 16911049 124422105 896158921
1 257 7327 105149 1137991 10657997 91989367 756570029 6046077511
1 513 21217 380745 4857001 52034913 504717697 4611314745 40608430681
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MAPLE
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sum((Stirling2(m, i)*factorial(i)+Stirling2(m, i+1)*factorial(i+1))*Stirling2(n, i), i = 1 .. n);
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MATHEMATICA
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Table[Sum[(StirlingS2[m, i] i! + StirlingS2[m, i + 1] (i + 1)!) StirlingS2[n - m + 1, i], {i, n - m + 1}], {n, 10}, {m, n, 1, -1}] // Flatten (* Michael De Vlieger, Dec 14 2015 *)
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PROG
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(PARI) T(n, m) = sum(i=1, n, ( stirling(m, i, 2)*i! + stirling(m, i+1, 2)*(i+1)!)*stirling(n, i, 2));
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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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