%I #35 Sep 08 2022 08:45:12

%S 1,4,24,200,2160,28812,458752,8503056,180000000,4287177620,

%T 113515167744,3308603804376,105288694411264,3632897460937500,

%U 135107988821114880,5388090449900829728,229385780960233586688,10383890888434362036516,498073600000000000000000

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

%C 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 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 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 Alois P. Heinz, <a href="/A089946/b089946.txt">Table of n, a(n) for n = 0..386</a>

%H 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 F. A. Haight, <a href="http://www.jstor.org/stable/2333538">Overflow at a traffic light</a>, Biometrika, 46 (1959), 420-424.

%H F. A. Haight, <a href="/A001787/a001787_3.pdf">Overflow at a traffic light</a>, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)

%H F. A. Haight, <a href="/A001787/a001787_2.pdf">Letter to N. J. A. Sloane, n.d.</a>

%F a(n) = 2*(n+1) * (n+2)^(n-1).

%F a(n) = Sum_{k=0..n} C(n, k) * (n-k+1)^(n-k-1) * (2*k+1) * (k+1)^(k-1).

%F E.g.f.: (-LambertW(x)/x)^2 * (1 - LambertW(x)) / (1 + LambertW(x)).

%t 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 Table[2*(n+1)*(n+2)^(n-1), {n, 0, 50}] (* _G. C. Greubel_, Nov 14 2017 *)

%o (PARI) a(n)=if(n<0,0,2*(n+1)*(n+2)^(n-1));

%o (Magma) [2*(n+1) * (n+2)^(n-1): n in [0..50]]; // _G. C. Greubel_, Nov 14 2017

%Y Cf. A089944, A089945.

%Y A diagonal of A259334.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 23 2003