A086331
Expansion of e.g.f. exp(x)/(1 + LambertW(-x)).
Original entry on oeis.org
1, 2, 7, 43, 393, 4721, 69853, 1225757, 24866481, 572410513, 14738647221, 419682895325, 13094075689225, 444198818128313, 16278315877572141, 640854237634448101, 26973655480577228769, 1208724395795734172705, 57453178877303382607717, 2887169565412587866031533
Offset: 0
a(2) = 7 because {}->{}, 1->1, 2->2, and the four functions from {1,2} into {1,2}. Note A000169(2) = 9 because it counts these 7 and 1->2, 2->1.
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a:= n-> add(binomial(n,k)*k^k, k=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Dec 30 2021
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nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[x]/(1-t),{x,0,nn}],x] (* Geoffrey Critzer, Dec 19 2011 *)
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a(n) = sum(k=0,n, binomial(n, k)*k^k ); \\ Joerg Arndt, May 10 2013
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022
Original entry on oeis.org
2, 3, 6, 13, 29, 71, 173, 444, 1148, 3030, 8059, 21715, 58836, 160687, 441083, 1217134, 3372386, 9380948, 26180962, 73292358, 205731862, 578922864, 1632707684, 4614098810, 13064064882, 37052720050, 105257568244, 299452309023, 853094139960, 2433439189419
Offset: 1
A127911
Number of nonisomorphic partial functional graphs with n points which are not functional graphs.
Original entry on oeis.org
0, 1, 3, 9, 26, 74, 208, 586, 1647, 4646, 13135, 37247, 105896, 301880, 862498, 2469480, 7083690, 20353886, 58571805, 168780848, 486958481, 1406524978, 4066735979, 11769294050, 34090034328, 98820719105, 286672555274
Offset: 0
a(0) = 0 because the null graph is trivially both partial functional and functional.
a(1) = 1 because there are two partial functional graphs on one point: the point with, or without, a loop; the point with loop is the identity function, but without a loop the naked point is the unique merely partial functional case.
a(2) = 3 because there are A126285(2) enumerates the 6 partial functional graphs on 2 points, of which 3 are functional, 6 - 3 = 3.
a(3) = A126285(3) - A001372(3) = 16 - 7 = 9.
a(4) = 45 - 19 = 26.
a(5) = 121 - 47 = 74.
a(6) = 338 - 130 = 208.
a(7) = 929 - 343 = 586.
a(8) = 2598 - 951 = 1647.
a(9) = 7261 - 2615 = 4646.
a(10) = 20453 - 7318 = 13135.
- S. Skiena, "Functional Graphs." Section 4.5.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 164-165, 1990.
A127912
Number of nonisomorphic disconnected mappings (or mapping patterns) from n points to themselves; number of disconnected endofunctions.
Original entry on oeis.org
0, 1, 3, 10, 27, 79, 218, 622, 1753, 5007, 14274, 40954, 117548, 338485, 975721, 2817871, 8146510, 23581381, 68322672, 198138512, 575058726, 1670250623, 4854444560, 14117859226, 41081418963, 119606139728
Offset: 0
a(0) = 0, as the null digraph is formally neither connected nor disconnected.
a(1) = 0, as the unique endofunction on one point is the identity function on one value and is connected.
a(2) = 1, as there are 3 endofunctions on two points, two of which are "prime endofunctions" and one of which is the direct sum of two copies of the unique endofunction on one point, namely two points-with-loops, or the identity function on two values; 3 - 2 = 1.
a(3) = A001372(3) - A002861(3) = 7 - 4 = 3.
a(4) = A001372(4) - A002861(4) = 19 - 9 = 10.
a(5) = A001372(5) - A002861(5) = 47 - 20 = 27.
a(6) = 130 - 51 = 79.
a(7) = 343 - 125 = 218.
a(8) = 951 - 329 = 622.
a(9) = 2615 - 862 = 1753.
a(10) = 7318 - 2311 = 5007.
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.6.
- R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.399 and 41.401.
A350571
Triangular array read by rows. T(n,k) is the number of unlabeled partial functions on [n] with exactly k undefined points, n>=0, 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 3, 2, 1, 7, 6, 2, 1, 19, 16, 7, 2, 1, 47, 45, 19, 7, 2, 1, 130, 121, 57, 20, 7, 2, 1, 343, 338, 158, 60, 20, 7, 2, 1, 951, 929, 457, 170, 61, 20, 7, 2, 1, 2615, 2598, 1286, 498, 173, 61, 20, 7, 2, 1, 7318, 7261, 3678, 1421, 510, 174, 61, 20, 7, 2, 1
Offset: 0
Triangle T(n,k) begins:
1;
1, 1;
3, 2, 1;
7, 6, 2, 1;
19, 16, 7, 2, 1;
47, 45, 19, 7, 2, 1;
130, 121, 57, 20, 7, 2, 1;
343, 338, 158, 60, 20, 7, 2, 1;
951, 929, 457, 170, 61, 20, 7, 2, 1;
...
- O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009.
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nn = 10; A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt",
"Table"], {, }][[;; nn, 2]];
A000081 = Drop[Cases[ Import["https://oeis.org/A000081/b000081.txt",
"Table"], {, }][[;; nn + 1, 2]], 1];
Map[Select[#, # > 0 &] &, CoefficientList[Series[ Product[1/(1 - y x^i)^A000081[[i]], {i, 1, nn}] Product[1/(1 - x^i)^A002861[[i]], {i, 1, nn}], {x, 0, nn}], {x,y}]] // Grid
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