José H. Nieto S. has authored 3 sequences.
A216779
Number of derangements on n elements with an odd number of cycles.
Original entry on oeis.org
0, 0, 1, 2, 6, 24, 135, 930, 7420, 66752, 667485, 7342290, 88107426, 1145396472, 16035550531, 240533257874, 3848532125880, 65425046139840, 1177650830516985, 22375365779822562, 447507315596451070, 9397653627525472280, 206748379805560389951, 4755212735527888968642
Offset: 0
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a := proc (n) local x, y, t, k; if n <= 1 then 0 else x := 0; y := 0; for k from 2 to n do t := y; y := (k-1)*(x+y-k+3); x := t end do; y end if end proc;
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nn=23;Range[0,nn]!CoefficientList[Series[Sinh[Log[1/(1-x)]-x],{x,0,nn}],x] (* Geoffrey Critzer, Jun 23 2014 *)
A216778
Number of derangements on n elements with an even number of cycles.
Original entry on oeis.org
1, 0, 0, 0, 3, 20, 130, 924, 7413, 66744, 667476, 7342280, 88107415, 1145396460, 16035550518, 240533257860, 3848532125865, 65425046139824, 1177650830516968, 22375365779822544, 447507315596451051, 9397653627525472260, 206748379805560389930, 4755212735527888968620
Offset: 0
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a := proc (n) local x, y, t, k; if n = 0 then 1 elif n = 1 then 0 else x := 1; y := 0; for k from 2 to n do t := y; y := (k-1)*(x+y+k-3); x := t end do; y end if end proc;
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nn=23;Range[0,nn]!*CoefficientList[Series[Cosh[Log[1/(1-x)]-x],{x,0,nn}],x] (* Geoffrey Critzer, Jun 23 2014 *)
A204420
Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.
Original entry on oeis.org
1, 0, 1, 0, 6, 3, 0, 120, 90, 15, 0, 5040, 4620, 1260, 105, 0, 362880, 378000, 132300, 18900, 945, 0, 39916800, 45571680, 18711000, 3534300, 311850, 10395, 0, 6227020800, 7628100480, 3511347840, 794593800, 94594500, 5675670, 135135
Offset: 0
1;
0, 1,
0, 6, 3;
0, 120, 90, 15;
0, 5040, 4620, 1260, 105;
0, 362880, 378000, 132300, 18900, 945;
0, 39916800, 45571680, 18711000, 3534300, 311850, 10395;
- Alois P. Heinz, Rows n = 0..85, flattened
- Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
- Angelo Lucia and Amanda Young, A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs, arXiv:2212.11872 [math-ph], 2022.
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T_row:= proc(n) local k; seq(doublefactorial(2*n-1)*2^(n-k)* coeff(expand(pochhammer(x, n)), x, k), k=0..n) end: seq(T_row(n), n=0..10);
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nn=12;Prepend[Map[Prepend[Select[#,#>0&],0]&,Table[(Range[0, nn]!CoefficientList[ Series[(1-x^2)^(-y/2),{x,0,nn}],{x,y}])[[n]],{n,3,nn,2}]],{1}]//Grid (* Geoffrey Critzer, Jul 21 2013 *)
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T(n,k) = (2*n)!/(2^n*n!)*(-2)^(n-k)*stirling(n,k,1); \\ Andrew Howroyd, Feb 12 2018
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