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A275364
Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes whose maximal connected component has at most k nodes, n>=1, 1<=k<=n.
1
1, 1, 2, 1, 4, 8, 1, 10, 26, 64, 1, 26, 106, 296, 1024, 1, 76, 556, 1696, 6064, 32768, 1, 232, 2752, 13392, 43968, 230896, 2097152, 1, 764, 15548, 135248, 461392, 1956816, 16886864, 268435456, 1, 2620, 99836, 1062224, 6932816, 24877904, 159248336, 2423185664, 68719476736, 1, 9496, 636056, 9621536, 130702496, 489604256, 2281210016, 24920583296, 687883494016, 35184372088832
OFFSET
1,3
LINKS
EXAMPLE
1,
1, 2,
1, 4, 8,
1, 10, 26, 64,
1, 26, 106, 296, 1024,
1, 76, 556, 1696, 6064, 32768,
1, 232, 2752, 13392, 43968, 230896, 2097152,
MAPLE
with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
add(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
end:
T:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
T(n-i*j, i-1)*b(i)^j, j=0..n/i)))
end:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Jul 26 2016
MATHEMATICA
nn = 10; f[z] := Sum[2^Binomial[n, 2] z^n/n!, {n, 0, nn}]; a = Drop[Range[0, nn]! CoefficientList[Series[Log[f[z]], {z, 0, nn}], z], 1]; Drop[Map[DeleteDuplicates, Transpose[Table[Range[0, nn]! CoefficientList[Series[Exp[Sum[a[[m]] z^m/m!, {m, 1, k}]], {z, 0, nn}], z], {k, 1, nn}]]], 1] // Grid
CROSSREFS
T(n,n) = A006125 for n>0.
T(n,2) = A000085 for n>1.
Cf. A001187.
Sequence in context: A221632 A071951 A264059 * A160323 A340469 A128411
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jul 24 2016
STATUS
approved