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%I #14 Jan 04 2021 19:24:08
%S 0,0,3,20,188,2280,33864,595196,12081600,278122032,7159299200,
%T 203771364324,6354217539072,215429796291320,7889813961243648,
%U 310413633428119500,13057068314325008384,584737112800511959104,27776659696045110558720,1395009275793285886030772,73854320834079368232960000
%N Number of functions f:{1,2,...,n} -> {1,2,...,n} such that the 1 and the 2 are in the same component of the functional digraph representation of f.
%F E.g.f. is the double integral of A''(x)*B(x) dx^2 where A(x) is the e.g.f. for A001865 and B(x) is the e.g.f. for A000312.
%e a(3)=20 because there are 17 connected functions on [3] and (2,1,3), (1,1,3), (2,2,3) where the functions are represented by their values.
%t nn=18; t=Sum[n^(n-1)x^n/n!, {n,1,nn+2}]; Join[{0,0}, Range[0,nn]! CoefficientList[Series[D[D[Log[1/(1-t)], x], x]/(1-t), {x,0,nn}], x]]
%t a[ n_] := If[ n < 2, 0, With[ {m = n - 2}, With[ {t = 1 + Sum[k^k x^k/k!, {k, m + 2}]}, m! SeriesCoefficient[ D[ Log[ t], {x, 2}] t, {x, 0, m} ]]]] (* _Michael Somos_, Jun 04 2013 *)
%o (PARI) {a(n) = local(A); if( n<2, 0, m = n-2; A = sum( k=0, m+2, k^k * x^k / k!, x^3 * O(x^m)); m! * polcoeff( log(A)'' * A, m))} /* _Michael Somos_, Jun 04 2013 */
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Jun 04 2013