A130639
Number of degree-2n permutations without even cycles and such that number of cycles of size 2k-1 is even (or zero) for every k.
Original entry on oeis.org
1, 1, 1, 41, 1121, 80977, 5073377, 984765497, 131026429249, 45819745767329, 9199822716980033, 5303459200225973833, 1646226697154555000993, 1377111876294420026771441, 574027598120143165861124641, 675477754387947155701063431257, 381022545331716847279242552317057
Offset: 0
a(2)=1 because we have (1)(2)(3)(4).
-
g:=product(cosh(x^(2*k-1)/(2*k-1)),k=1..40): gser:=series(g,x=0,35): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..14); # Emeric Deutsch, Aug 25 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
`if`(j=0 or irem(i, 2)=1 and irem(j, 2)=0, multinomial(n,
n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Mar 09 2015
-
multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[j == 0 || Mod[i, 2] == 1 && Mod[j, 2] == 0, multinomial[n, Join[{n-i*j}, Array[i&, j]]]*(i-1)!^j/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]]; a[n_] := b[2n, 2n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)
A130644
Number of degree-2n permutations without odd cycles and such that number of cycles of size 2k is odd (or zero) for every k.
Original entry on oeis.org
1, 1, 6, 225, 8400, 760725, 91725480, 15563633085, 3381661483200, 1015992072520425, 360153767651277600, 160068908768727783825, 84298688029883001074400, 53051020433282263735468125, 38316864396320965168213500000, 32660810942813910822645908353125
Offset: 0
a(2)=6 because we have (1234),(1243),(1324),(1342),(1423) and (1432).
-
g:=product(1+sinh(x^(2*k)/(2*k)),k=1..50): gser:=series(g,x=0,44): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..14); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
`if`(j=0 or irem(i, 2)=0 and irem(j, 2)=1, multinomial(n,
n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Mar 09 2015
-
multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 && Mod[j, 2] == 1, multinomial[n, Join[{n-i*j}, Array[i&, j]]]*(i-1)!^j/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]]; a[n_] := b[2n, 2n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)
A130648
Number of degree-n permutations without even cycles and such that number of cycles of size 2k-1 is odd (or zero) for every k.
Original entry on oeis.org
1, 1, 0, 3, 8, 25, 184, 721, 9904, 66753, 691088, 5973121, 84925048, 940427137, 12801319816, 186556383105, 3174772979936, 48489077948161, 842173637012896, 15359492773456129, 316965131969908072, 6368424993521096961, 135098381153771956952, 2980219360336428021505
Offset: 0
a(3)=3 because we have (1)(2)(3), (123) and (132).
-
g:=product(1+sinh(x^(2*k-1)/(2*k-1)),k=1..30): gser:=series(g,x=0,27): seq(factorial(n)*coeff(gser,x,n),n=0..24); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
`if`(j=0 or irem(i, 2)=1 and irem(j, 2)=1, multinomial(n,
n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 09 2015
-
multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 1 && Mod[j, 2] == 1, multinomial[n, Join[{n-i*j}, Array[i&, j]]]*(i-1)!^j/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)
A131526
Number of degree-n permutations such that number of cycles of size 2k is even (or zero) and number of cycles of size 2k-1 is odd (or zero), for every k.
Original entry on oeis.org
1, 1, 0, 3, 11, 40, 184, 1036, 12949, 88488, 807008, 7362586, 113572183, 1238477032, 15630890560, 228998728050, 4141605806441, 62222251093216, 1030119451142656, 19050688698470434, 412037845709792107, 8102391640556570616, 165794307361686866432
Offset: 0
a(4)=11 because we have (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3), (143)(2), (12)(34), (13)(24) and (14)(23).
-
g:=(product(1+sinh(x^(2*k-1)/(2*k-1)), k=1..40))*(product(cosh(x^(2*k)/(2*k)), k=1..40)): gser:=series(g,x=0,25); seq(factorial(n)*coeff(gser,x,n),n=0..21); # Emeric Deutsch, Aug 28 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
`if`(j=0 or irem(i+j, 2)=0, multinomial(n, n-i*j, i$j)*
(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 09 2015
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i + j, 2] == 0, multinomial[n, {n - i j} ~Join~ Table[i, {j}]] (i - 1)!^j/j! b[n - i j, i - 1], 0], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 19 2020, after Alois P. Heinz *)
Showing 1-4 of 4 results.