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A136394
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Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).
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14
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1, 1, 1, 1, 1, 5, 1, 20, 3, 1, 84, 35, 1, 409, 295, 15, 1, 2365, 2359, 315, 1, 16064, 19670, 4480, 105, 1, 125664, 177078, 56672, 3465, 1, 1112073, 1738326, 703430, 74025, 945, 1, 10976173, 18607446, 8941790, 1346345, 45045, 1, 119481284, 216400569, 118685336
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OFFSET
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0,6
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LINKS
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FORMULA
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E.g.f.: exp(x*(1-y))/(1-x)^y. Binomial transform of triangle A008306. exp(x)*((-x-log(1-x))^k)/k! is e.g.f. of k-th column.
Sum_{k=0..floor(n/2)} k * T(n,k) = A001705(n-1) for n>=1.
Sum_{k=0..floor(n/2)} (-1)^k * T(n,k) = A159964(n-1) for n>=1. (End)
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EXAMPLE
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Triangle (n,k) begins:
1;
1;
1, 1;
1, 5;
1, 20, 3;
1, 84, 35;
1, 409, 295, 15;
1, 2365, 2359, 315;
...
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MAPLE
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egf:= proc(k::nonnegint) option remember; x-> exp(x)* ((-x-ln(1-x))^k)/k! end; T:= (n, k)-> coeff(series(egf(k)(x), x=0, n+1), x, n) *n!; seq(seq(T(n, k), k=0..n/2), n=0..30); # Alois P. Heinz, Aug 14 2008
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
`if`(i>1, x, 1)*binomial(n-1, i-1)*(i-1)!, i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
# third Maple program:
T:= proc(n, k) option remember; `if`(k<0 or k>2*n, 0,
`if`(n=0, 1, add(T(n-i, k-`if`(i>1, 1, 0))*
mul(n-j, j=1..i-1), i=1..n)))
end:
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MATHEMATICA
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max = 12; egf = Exp[x*(1-y)]/(1-x)^y; s = Series[egf, {x, 0, max}, {y, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]*n!; t[0, 0] = t[1, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)
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CROSSREFS
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Columns k=0-10 give: A000012, A006231, A289950, A289951, A289952, A289953, A289954, A289955, A289956, A289957, A289958.
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KEYWORD
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AUTHOR
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STATUS
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approved
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