cp's OEIS Frontend

A275707 Number of partial functions f:{1,2,...,n}->{1,2,...,n} such that every element in the domain of definition of f is mapped to a fixed point or to an element that is undefined by f.

%I A275707 #44 Jul 06 2022 04:30:30
%S A275707 1,2,8,38,216,1402,10156,80838,698704,6498674,64579284,681642238,
%T A275707 7605025720,89318058858,1100376445564,14176837311158,190498308591264,
%U A275707 2663482511782114,38667106019619748,581765160424218606,9055862445043643656,145619330650420134362
%N A275707 Number of partial functions f:{1,2,...,n}->{1,2,...,n} such that every element in the domain of definition of f is mapped to a fixed point or to an element that is undefined by f.
%H A275707 Alois P. Heinz, <a href="/A275707/b275707.txt">Table of n, a(n) for n = 0..524</a>
%F A275707 E.g.f.: A(x)^2 = exp(2*B(x)) where A(x) is the e.g.f. for A000248 and B(x) is the e.g.f. for A000027.
%F A275707 E.g.f.: exp(2*x*exp(x)). - _Joerg Arndt_, Nov 10 2016
%F A275707 a(0) = 1; a(n) = Sum_{k=1..n} 2*k*binomial(n-1,k-1)*a(n-k). - _Ilya Gutkovskiy_, Nov 24 2017
%F A275707 From _Seiichi Manyama_, Jul 04 2022: (Start)
%F A275707 G.f.: Sum_{k>=0} (2 * x)^k/(1 - k*x)^(k+1).
%F A275707 a(n) = Sum_{k=0..n} 2^k * k^(n-k) * binomial(n,k). (End)
%F A275707 a(n) ~ n^(n + 1/2) * exp(2*r*exp(r) - r/2 - n) / (sqrt(2*(1 + 3*r + r^2)) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n)/2^(3/2)). - _Vaclav Kotesovec_, Jul 06 2022
%e A275707 G.f. = 1 + 2*x + 8*x^2 + 38*x^3 + 216*x^4 + 1402*x^5 + 10156*x^6 + ...
%e A275707 a(2) = 8 because there are 9 = A000169(3) partial functions on a set with 2 elements and all of them have the stated property except 1->2,2->1.
%p A275707 a:= n-> add(binomial(n, k)*add(binomial(n-k, f)*
%p A275707         (f+k)^(n-k-f), f=0..n-k), k=0..n):
%p A275707 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 07 2016
%t A275707 nn = 20; Range[0, nn]! CoefficientList[Series[ Exp[z Exp[z]]^2, {z, 0, nn}], z]
%t A275707 Table[Sum[BellY[n, k, 2 Range[n]], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o A275707 (PARI) x='x+O('x^33); Vec(serlaplace(exp(2*x*exp(x)))) \\ _Joerg Arndt_, Nov 10 2016
%o A275707 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*x)^k/(1-k*x)^(k+1))) \\ _Seiichi Manyama_, Jul 04 2022
%o A275707 (PARI) a(n) = sum(k=0, n, 2^k*k^(n-k)*binomial(n, k)); \\ _Seiichi Manyama_, Jul 04 2022
%Y A275707 Column 2 of A187105.
%Y A275707 Cf. A000027, A000169, A000248, A351762, A355501.
%K A275707 nonn
%O A275707 0,2
%A A275707 _Geoffrey Critzer_, Aug 06 2016