A096939
Number of sets of odd number of even lists, cf. A000262.
Original entry on oeis.org
0, 2, 6, 36, 260, 1950, 19362, 193256, 2326536, 29272410, 413257790, 6231230412, 101415565836, 1769925341366, 32734873484250, 646218442877520, 13404753632014352, 294656673023216946, 6775966692145553526
Offset: 1
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Drop[ Range[0, 20]! CoefficientList[ Series[ Exp[(x/(1 - x^2))] Sinh[x^2/(1 - x^2)], {x, 0, 20}], x], 1] (* Robert G. Wilson v, Aug 19 2004 *)
A349776
Triangle read by rows: T(n,k) is the number of partitions of set [n] into a set of at most k lists, with 0 <= k <= n. Also called broken permutations.
Original entry on oeis.org
1, 0, 1, 0, 2, 3, 0, 6, 12, 13, 0, 24, 60, 72, 73, 0, 120, 360, 480, 500, 501, 0, 720, 2520, 3720, 4020, 4050, 4051, 0, 5040, 20160, 32760, 36960, 37590, 37632, 37633, 0, 40320, 181440, 322560, 381360, 393120, 394296, 394352, 394353
Offset: 0
For n=3 the T(3, 2)=12 broken permutations are {(1, 2, 3)}, {(1, 3, 2)}, {(2, 1, 3)}, {(2, 3, 1)}, {(3, 1, 2)}, {(3, 2, 1)}, {(1, 2), (3)}, {(2, 1), (3)}, {(1, 3), (2)}, {(3, 1), (2)}, {(2, 3), (1)}, {(3, 2), (1)}.
If you add the set of 3 lists {(1), (2), (3)}, you get T(3, 3) = 13 = A000262(3).
Triangle begins:
1;
0, 1;
0, 2, 3;
0, 6, 12, 13;
0, 24, 60, 72, 73;
0, 120, 360, 480, 500, 501;
0, 720, 2520, 3720, 4020, 4050, 4051;
...
- Kenneth P. Bogart, Combinatorics Through Guided Discovery, Kenneth P. Bogart, 2004, 57-58.
Row sums give
A062147(n-1) for n>=1.
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T:= proc(n, k) option remember; `if`(k<0, 0,
binomial(n-1, k-1)*n!/k! +T(n, k-1))
end:
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Dec 01 2021
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T[n_, k_] := Sum[Binomial[n-1, n-j] n!/j!, {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2023 *)
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Lah(n, k) = if (n==0, 1, binomial(n-1, k-1)*n!/k!); \\ A271703
T(n, k) = sum(j=0, k, Lah(n, j)); \\ Michel Marcus, Nov 30 2021
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def T(n, k):
return sum(binomial(n, i)*falling_factorial(n-1, n-i) for i in (0..k))
print([[T(n, k) for k in (0..n)] for n in (0..9)]) # Peter Luschny, Dec 01 2021
Showing 1-2 of 2 results.
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