A092691
a(n) = n! * Sum_{k=1..floor(n/2)} 1/(2k).
Original entry on oeis.org
0, 0, 1, 3, 18, 90, 660, 4620, 42000, 378000, 4142880, 45571680, 586776960, 7628100480, 113020427520, 1695306412800, 28432576972800, 483353808537600, 9056055981772800, 172065063653683200, 3562946373482496000, 74821873843132416000, 1697172166720622592000
Offset: 0
a(4)=4!*(1/2+1/4)=18, a(5)=5!*(1/2+1/4)=90.
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 3.3.13.
-
nn = 20; Range[0, nn]! CoefficientList[
D[Series[(1 - x^2)^(-y/2) ((1 + x)/(1 - x))^(1/2), {x, 0, nn}], y] /. y -> 1, x] (* Geoffrey Critzer, Aug 27 2012 *)
-
a(n)=if(n<0,0,n!*sum(k=1,n\2,1/k)/2)
-
{a(n)=if(n<0, 0, n!*polcoeff( log(1-x^2+x*O(x^n))/(2*x-2), n))}
A151882
Let g be a permutation of [1..n] having say j_i cycles of length i, with Sum_i i*j_i = n; sequence gives Sum_g Sum_i (j_i)^2.
Original entry on oeis.org
1, 5, 17, 80, 424, 2744, 19928, 166984, 1543176, 15939792, 178966512, 2200820544, 29089668672, 415261531008, 6316101256320, 102692213061120, 1766690411927040, 32235156493470720, 618870347081671680, 12523381062124032000, 265423904312781312000
Offset: 1
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, [1,0], `if`(i<1, 0,
add(multinomial(n,n-i*j,i$j)/j!*(i-1)!^j*(p-> p+
[0, p[1]*j^2])(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30); # Alois P. Heinz, Oct 21 2015
-
multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]] ]/j! * (i-1)!^j*Function[p, p+{0, p[[1]]*j^2}][b[n-i*j, i-1]], {j, 0, n/i}] ] ]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2017, after Alois P. Heinz *)
A151884
Let g be a permutation of [1..n] having say j_i cycles of length i, with Sum_i i*j_i = n; sequence gives Sum_g Sum_{i odd} (j_i)^2.
Original entry on oeis.org
1, 4, 14, 56, 304, 1904, 14048, 112384, 1051776, 10662912, 120920832, 1451049984, 19342651392, 272576268288, 4175822315520, 66813157048320, 1156746459709440, 20900477925457920, 403511454289428480, 8070229085788569600, 171907712809736601600
Offset: 1
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, [1,0], `if`(i<1, 0,
add(multinomial(n,n-i*j,i$j)/j!*(i-1)!^j*(p-> p+
`if`(i::odd, [0, p[1]*j^2], 0))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30); # Alois P. Heinz, Oct 21 2015
-
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*(i-1)!^j*Function[p, p+If[OddQ[i], {0, p[[1]]*j^2}, {0, 0}]][b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 13 2017, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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