A201720
The total number of components in (A011800) of all labeled forests on n nodes whose components are all paths.
Original entry on oeis.org
0, 1, 3, 12, 64, 420, 3246, 28798, 288072, 3205044, 39234340, 523821936, 7572221328, 117792884872, 1961516974704, 34807390821960, 655594811020096, 13060711726818768, 274358217793164912, 6060159633360214144, 140404595387426964480
Offset: 0
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A201720 := proc(n)
g := (2*x-x^2)*exp((2*x-x^2)/(2-2*x))/(2-2*x) ;
coeftayl(g,x=0,n) ;
%*n! ;
end proc:
seq(A201720(n),n=0..30) ; # R. J. Mathar, Jun 27 2022
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D[Range[0, 20]! CoefficientList[ Series[Exp[y (2 x - x^2)/(2 - 2 x)], {x, 0, 20}], x], y] /. y -> 1
A330021
Expansion of e.g.f. exp(sinh(exp(x) - 1)).
Original entry on oeis.org
1, 1, 2, 6, 25, 128, 754, 5001, 37048, 303930, 2732395, 26657106, 280039786, 3149224991, 37729906686, 479570263690, 6442902231289, 91186621152460, 1355582225366134, 21112253012491481, 343672026658191836, 5834977672879651390, 103130592695715620419
Offset: 0
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g:= proc(n) option remember; `if`(n=0, 1, add(
binomial(n-1, j-1)*irem(j, 2)*g(n-j), j=1..n))
end:
b:= proc(n, m) option remember; `if`(n=0,
g(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..22); # Alois P. Heinz, Jun 23 2023
-
nmax = 22; CoefficientList[Series[Exp[Sinh[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!
A202081
The number of simple labeled graphs on n nodes whose connected components are cycles, stars, wheels, or paths.
Original entry on oeis.org
1, 1, 2, 8, 46, 298, 2206, 19009, 187076, 2053349, 24800484, 327067043, 4677505768, 72075818159, 1189985755128, 20952274850927, 391829421176768, 7755079821666945, 161926610838369418, 3556807008080385549, 81979632030102053376, 1978135038931568355707
Offset: 0
- R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge 1999, problem 5.15
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nn = 16; a = x/(2 (1 - x)) + x/2; b = x^4/4! + Sum[(n (n - 2)!/2) x^n/n!, {n, 5, nn}]; c = x Exp[x] - x^3/2 - x^2 - x; d = -x/2 - x^2/4; Range[0, nn]! CoefficientList[Series[Exp[a]*Exp[b]*Exp[c]*Exp[d]/(1 - x)^(1/2), {x, 0, nn}], x]
A275488
Number of labeled forests of (free) trees such that exactly one tree is a path.
Original entry on oeis.org
1, 1, 3, 12, 80, 810, 10857, 174944, 3243060, 67859010, 1586109305, 41085509652, 1170954002946, 36469499267474, 1233416773419495, 45037748851872240, 1766375778253548392, 74067278799492363330, 3306928891056821667045, 156635771633727023132300
Offset: 1
a(1),a(2),a(3),a(4) are just a single path through an empty forest. a(5)=80 counts the 60 labelings of a path on 5 nodes and the 20 labelings of a path on 1 node and a star on 4 nodes.
- J. Harris, J. Hirst, M. Mossinghoff, Combinatorics and Graph Theory, Springer, 2010, page 34.
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nn = 20; b[z_] := 1/((1 - z) 2) - 1/2 + z/2;
t[z_] := z + Sum[n^(n - 2) z^n/n!, {n, 2, nn}];
Drop[Range[0, nn]! CoefficientList[Series[b[z] Exp[t[z] - b[z]], {z, 0, nn}], z], 1]
Showing 1-4 of 4 results.
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