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A285439
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Sum T(n,k) of the entries in the k-th cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
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8
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1, 4, 2, 21, 12, 3, 132, 76, 28, 4, 960, 545, 235, 55, 5, 7920, 4422, 2064, 612, 96, 6, 73080, 40194, 19607, 6692, 1386, 154, 7, 745920, 405072, 202792, 75944, 18736, 2816, 232, 8, 8346240, 4484808, 2280834, 911637, 254061, 46422, 5256, 333, 9
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OFFSET
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1,2
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COMMENTS
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Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
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LINKS
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FORMULA
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Sum_{k=1..n} k * T(n,k) = n^2 * n! = A002775(n).
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EXAMPLE
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T(3,1) = 21 because the sum of the entries in the first cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 6+6+3+4+1+1 = 21.
Triangle T(n,k) begins:
1;
4, 2;
21, 12, 3;
132, 76, 28, 4;
960, 545, 235, 55, 5;
7920, 4422, 2064, 612, 96, 6;
73080, 40194, 19607, 6692, 1386, 154, 7;
745920, 405072, 202792, 75944, 18736, 2816, 232, 8;
...
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MAPLE
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T:= proc(h) option remember; local b; b:=
proc(n, l) option remember; `if`(n=0, [mul((i-1)!, i=l), 0],
(p-> p+[0, (h-n+1)*p[1]*x^(nops(l)+1)])(b(n-1, [l[], 1]))+
add((p-> p+[0, (h-n+1)*p[1]*x^j])(
b(n-1, subsop(j=l[j]+1, l))), j=1..nops(l)))
end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2])
end:
seq(T(n), n=1..10);
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MATHEMATICA
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T[h_] := T[h] = Module[{b}, b[n_, l_] := b[n, l] = If[n == 0, {Product[(i - 1)!, {i, l}], 0}, # + {0, (h - n + 1)*#[[1]]*x^(Length[l] + 1)}&[b[n - 1, Append[l, 1]]] + Sum[# + {0, (h-n+1)*#[[1]]*x^j}&[b[n - 1, ReplacePart[ l, j -> l[[j]] + 1]]], {j, 1, Length[l]}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, {}][[2]]]];
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CROSSREFS
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Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
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KEYWORD
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AUTHOR
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STATUS
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approved
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