A243098
Number T(n,k) of endofunctions on [n] with all cycles of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 16, 6, 2, 0, 125, 51, 24, 6, 0, 1296, 560, 300, 120, 24, 0, 16807, 7575, 4360, 2160, 720, 120, 0, 262144, 122052, 73710, 41160, 17640, 5040, 720, 0, 4782969, 2285353, 1430016, 861420, 430080, 161280, 40320, 5040
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 16, 6, 2;
0, 125, 51, 24, 6;
0, 1296, 560, 300, 120, 24;
0, 16807, 7575, 4360, 2160, 720, 120;
0, 262144, 122052, 73710, 41160, 17640, 5040, 720;
...
Main diagonal gives
A000142(n-1) for n>0.
-
with(combinat):
T:= (n, k)-> `if`(k*n=0, `if`(k+n=0, 1, 0),
add(binomial(n-1, j*k-1)*n^(n-j*k)*(k-1)!^j*
multinomial(j*k, k$j, 0)/j!, j=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..10);
-
multinomial[n_, k_] := n!/Times @@ (k!); T[n_, k_] := If[k*n==0, If[k+n == 0, 1, 0], Sum[Binomial[n-1, j*k-1]*n^(n-j*k)*(k-1)!^j*multinomial[j*k, Append[Array[k&, j], 0]]/j!, {j, 0, n/k}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)
A060917
Expansion of e.g.f.: exp((-1)^k/k*LambertW(-x)^k)/(k-1)!, k=3.
Original entry on oeis.org
1, 12, 150, 2180, 36855, 715008, 15697948, 385300800, 10463945085, 311697869120, 10108450408914, 354630018043392, 13384651003544275, 540860323696035840, 23300648262667635960, 1066165291831917811712
Offset: 3
-
nn = 20; CoefficientList[Series[E^(-1/3*LambertW[-x]^3)/2, {x, 0, nn}], x]* Range[0, nn]! (* Vaclav Kotesovec, Nov 27 2012 *)
-
x='x+O('x^30); Vec(serlaplace(exp(-lambertw(-x)^3/3)/2 - 1/2)) \\ G. C. Greubel, Feb 19 2018
A060918
Expansion of e.g.f.: exp((-1)^k/k*LambertW(-x)^k)/(k-1)!, k=4.
Original entry on oeis.org
1, 20, 360, 6860, 143570, 3321864, 84756000, 2372001720, 72384192540, 2394775746220, 85443353291296, 3271908306712500, 133893717061821080, 5832748749666611920, 269542701201588099840, 13172225935626444660144, 678788199609330554538000, 36790272488566573278647940
Offset: 4
-
CoefficientList[Series[E^(1/4*LambertW[-x]^4)/6, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
-
x='x+O('x^30); Vec(serlaplace(exp(lambertw(-x)^4/4)/3! - 1/3!)) \\ G. C. Greubel, Feb 19 2018
A332237
E.g.f.: -log(1 + LambertW(-x) * (2 + LambertW(-x)) / 2).
Original entry on oeis.org
1, 2, 8, 49, 409, 4356, 56734, 877094, 15742521, 322454800, 7434673036, 190792267128, 5398552673617, 167087263076384, 5617979017621650, 203987454978218416, 7957053981454827601, 331920300203780633856, 14746208516909980554736, 695208730205550274544000
Offset: 1
-
nmax = 20; CoefficientList[Series[-Log[1 + LambertW[-x] (2 + LambertW[-x])/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := a[n] = n^(n - 2) + (1/n) Sum[Binomial[n, k] (n - k)^(n - k - 2) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]
A072962
Unsigned reduced Euler characteristic for the matroid complex of cycle matroid for complete bipartite graph K_{n,n}.
Original entry on oeis.org
1, 20, 1071, 107104, 17201225, 4053135456, 1318104508735, 565989104282624, 310299479406324369, 211554189796535488000, 175592153482084893991151, 174356954302176729972264960, 204111110614488911169799727641, 278218647289052493421682954399744
Offset: 2
W. Kook and L. Thoma (andrewk(AT)math.uri.edu), Aug 20 2002
a(2,2)=1. Since K_{2,2} is a cycle with four edges, the matroid complex of cycle matroid for K_{2,2} is the 2-skeleton of standard 3-simplex. Therefore the unsigned reduced Euler characteristic for this complex is |-1+4-6+4|=1
- I. Novik, A. Postnikov and B. Sturmfels: Syzygies of oriented matroids, Duke Math. J. 111 (2002), no. 2, 287-317.
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