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Revision History for A224510

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A224510

Number of simple labeled graphs on {1,2,...,n} such that the node labeled with 1 is in the same component as the node labeled with 2.
(history; published version)

#22 by Joerg Arndt at Wed Apr 10 01:47:08 EDT 2013

STATUS

proposed

approved

#21 by Geoffrey Critzer at Tue Apr 09 13:08:30 EDT 2013

STATUS

editing

proposed

#20 by Alois P. Heinz at Tue Apr 09 12:03:30 EDT 2013

FORMULA

E.g.f.: Double integral of A''(x)*B(x) dx ^2 where A(x) is e.g.f . for A001187 and B(x) is e.g.f. for A006125. - _Geoffrey Critzer_, Apr 09 2013

Discussion

Tue Apr 09

13:08

Geoffrey Critzer: Alois Heinz

#19 by Alois P. Heinz at Tue Apr 09 11:59:43 EDT 2013

COMMENTS

A001187(n) < a(n) < A006125(n) for n>2.

STATUS

reviewed

editing

#18 by Joerg Arndt at Tue Apr 09 11:05:01 EDT 2013

STATUS

proposed

reviewed

#17 by Geoffrey Critzer at Tue Apr 09 08:27:39 EDT 2013

STATUS

editing

proposed

Discussion

Tue Apr 09

11:05

Joerg Arndt: Out of curiosity: who gave the formula?

#16 by Geoffrey Critzer at Tue Apr 09 08:24:37 EDT 2013

FORMULA

E.g.f.: Double integral of A''(x)*B(x)dx where A(x) is e.g.f for A001187 and B(x) is e.g.f. for A006125.

MATHEMATICA

(* by brute force counting *) nn=10; g=Sum[2^Binomial[n, 2]x^n/n!, {n, 0, nn}]; a=Drop[Range[0, nn]!CoefficientList[Series[Log[g]+1, {x, 0, nn}], x], 1]; f[list_]:=Product[a[[i]], {i, list}]; Table[Total[Map[f, Map[Length, Select[SetPartitions[n], MemberQ[#[[1]], 2]&], {2}]]], {n, 2, nn}]

(* or *)

nn=30; g=Sum[2^Binomial[n, 2]x^n/n!, {n, 0, nn+2}]; Range[0, nn]!CoefficientList[Series[D[D[Log[g]+1, x], x] g, {x, 0, nn}], x]

STATUS

approved

editing

Discussion

Tue Apr 09

08:27

Geoffrey Critzer: Thank you very much for the nice formula!

#15 by Alois P. Heinz at Tue Apr 09 08:16:50 EDT 2013

STATUS

editing

approved

#14 by Alois P. Heinz at Tue Apr 09 08:16:21 EDT 2013

MAPLE

a:= n-> add(binomial(n-2, k)*b(k+2)*2^(binomial(n-k-2)*(n-k-3)/, 2), k=0..n-2):

STATUS

reviewed

editing

#13 by Joerg Arndt at Tue Apr 09 07:42:34 EDT 2013

STATUS

proposed

reviewed