OFFSET
0,8
COMMENTS
Row sums = A001710(n).
If a permutation of {1, 2, ..., n} is written as a product of m disjoint cycles (where the fixed points of the permutation are viewed as 1-cycles) then the parity of the permutation is (-1)^(n-m). It is an even permutation if the number of cycles of even length is even (possibly zero), and it is an odd permutation if the number of cycles of even length is odd. - Peter Bala, Jun 25 2024
REFERENCES
J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 87, problem # 20.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Parity of a permutaion
FORMULA
E.g.f.: exp(y*A(x))*cosh(y*B(x)) where A(x)= log((1 + x)/(1 - x))^(1/2) and B(x)=log(1/(1-x^2)^(1/2).
From Peter Bala, Jun 25 2024: (Start)
If n and k are both even or both odd, then T(n, k) is equal to the Stirling cycle number |s(n, k)| = A132393(n, k), and 0 otherwise.
n-th row polynomial R(n, x) = (1/2)*( x*(x + 1)*...*(x + n + 1) + x*(x - 1)*...*(x - n - 1) ).
For n >= 1, the zeros of R(n, x) are purely imaginary. (End)
EXAMPLE
1,
0, 1,
0, 0, 1,
0, 2, 0, 1,
0, 0, 11, 0, 1,
0, 24, 0, 35, 0, 1,
0, 0, 274, 0, 85, 0, 1,
0, 720, 0, 1624, 0, 175, 0, 1,
0, 0, 13068, 0, 6769, 0, 322, 0, 1,
0, 40320, 0, 118124, 0, 22449, 0, 546, 0, 1,
0, 0, 1026576, 0, 723680, 0, 63273, 0, 870, 0, 1
MAPLE
with(combinat):
b:= proc(n, i, t) option remember; expand(`if`(n=0, t, `if`(i<1,
0, add(x^j*multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j,
i-1, irem(t+`if`(irem(i, 2)=0, j, 0), 2)), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 1)):
seq(T(n), n=0..12); # Alois P. Heinz, Mar 09 2015
# Alternative:
A132393 := (n, k) -> abs(Stirling1(n, k)):
T := (n, k) -> ifelse((n::even and k::even) or (n::odd and k::odd), A132393(n, k),
0): seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Jun 26 2024
MATHEMATICA
nn=11; a=Log[((1+x)/(1-x))^(1/2)]; b=Log[1/(1-x^2)^(1/2)]; Table[Take[(Range[0, nn]!CoefficientList[Series[Exp[y a]Cosh[y b] , {x, 0, nn}], {x, y}])[[n]], n], {n, 1, nn}]//Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Feb 16 2014
STATUS
approved