|
Search: id:A114320
|
|
|
| A114320 |
|
Triangle T(n,k) = number of permutations of n elements with k 2-cycles. |
|
+0
1
|
|
| 1, 1, 1, 3, 3, 15, 6, 3, 75, 30, 15, 435, 225, 45, 15, 3045, 1575, 315, 105, 24465, 12180, 3150, 420, 105, 220185, 109620, 28350, 3780, 945, 2200905, 1100925, 274050, 47250, 4725, 945, 24209955, 12110175, 3014550, 519750, 51975, 10395, 290529855
(list; graph; listen)
|
|
|
OFFSET
|
0,4
|
|
|
COMMENT
|
Row n has 1+floor(n/2) terms. Row sums yield the factorials (A000142). Sum(k*T(n,k),k>0)=n!/2 for n>=2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
|
|
FORMULA
|
E.g.f.: exp((y-1)*x^2/2)/(1-x). More generally, e.g.f. for number of permutations of n elements with k m-cycles is exp((y-1)*x^m/m)/(1-x).
|
|
EXAMPLE
|
1; 1; 1,1; 3,3; 15,6,3; 75,30,15; 435,225,45,15; ...
T(3,1)=3 because we have (1)(23), (12)(3) and (13)(2).
|
|
MAPLE
|
G:=exp((y-1)*x^2/2)/(1-x): Gser:=simplify(series(G, x=0, 15)): P[0]:=1: for n from 1 to 12 do P[n]:=n!*coeff(Gser, x^n) od: for n from 0 to 12 do seq(coeff(y*P[n], y^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
|
|
CROSSREFS
|
Cf. A008290, A000266, A000090, A088436, A000138, A060725, A060726, A086659, A105422, A105114.
Sequence in context: A100347 A165405 A163590 this_sequence A160612 A086116 A100735
Adjacent sequences: A114317 A114318 A114319 this_sequence A114321 A114322 A114323
|
|
KEYWORD
|
easy,nonn,tabf
|
|
AUTHOR
|
Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 05 2006
|
|
EXTENSIONS
|
More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
|
|
|
Search completed in 0.003 seconds
|
