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A089946 Secondary diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

%I A089946 #35 Sep 08 2022 08:45:12
%S A089946 1,4,24,200,2160,28812,458752,8503056,180000000,4287177620,
%T A089946 113515167744,3308603804376,105288694411264,3632897460937500,
%U A089946 135107988821114880,5388090449900829728,229385780960233586688,10383890888434362036516,498073600000000000000000
%N A089946 Secondary diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.
%C A089946 Also the hyperbinomial transform of A089945 (the main diagonal of A089944): a(n) = Sum_{k=0..n} C(n,k)*(n-k+1)^(n-k-1)*A089945(k).
%C A089946 With offset 1, a(n) = total number of children of the root in all (n+1)^(n-1) trees on {0,1,2,...,n} rooted at 0. For example, with edges directed away from the root, the trees on {0,1,2} are {0->1,0->2},{0->1->2},{0->2->1} and contain a total of a(2)=4 children of 0. - _David Callan_, Feb 01 2007
%C A089946 With offset 1, a(n) is the number of labeled rooted trees in all rooted forests on n nodes. The E.g.f. is B(T(x)) where  B(x)=x*exp(x) and T(x) is Euler's tree function. - _Geoffrey Critzer_, Oct 07 2011
%H A089946 Alois P. Heinz, <a href="/A089946/b089946.txt">Table of n, a(n) for n = 0..386</a>
%H A089946 Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H A089946 F. A. Haight, <a href="http://www.jstor.org/stable/2333538">Overflow at a traffic light</a>, Biometrika, 46 (1959), 420-424.
%H A089946 F. A. Haight, <a href="/A001787/a001787_3.pdf">Overflow at a traffic light</a>, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
%H A089946 F. A. Haight, <a href="/A001787/a001787_2.pdf">Letter to N. J. A. Sloane, n.d.</a>
%F A089946 a(n) = 2*(n+1) * (n+2)^(n-1).
%F A089946 a(n) = Sum_{k=0..n} C(n, k) * (n-k+1)^(n-k-1) * (2*k+1) * (k+1)^(k-1).
%F A089946 E.g.f.: (-LambertW(x)/x)^2 * (1 - LambertW(x)) / (1 + LambertW(x)).
%t A089946 t=Sum[n^(n-1)x^n/n!, {n,1,20}]; Drop[Range[0,20]!*CoefficientList[ Series[t*Exp[t], {x,0,20}], x], 1] (* _Geoffrey Critzer_, Oct 07 2011 *)
%t A089946 Table[2*(n+1)*(n+2)^(n-1), {n, 0, 50}] (* _G. C. Greubel_, Nov 14 2017 *)
%o A089946 (PARI) a(n)=if(n<0,0,2*(n+1)*(n+2)^(n-1));
%o A089946 (Magma) [2*(n+1) * (n+2)^(n-1): n in [0..50]]; // _G. C. Greubel_, Nov 14 2017
%Y A089946 Cf. A089944, A089945.
%Y A089946 A diagonal of A259334.
%K A089946 nonn
%O A089946 0,2
%A A089946 _Paul D. Hanna_, Nov 23 2003