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

1, 4, 24, 200, 2160, 28812, 458752, 8503056, 180000000, 4287177620, 113515167744, 3308603804376, 105288694411264, 3632897460937500, 135107988821114880, 5388090449900829728, 229385780960233586688, 10383890888434362036516, 498073600000000000000000

Offset: 0

Author

Paul D. Hanna, Nov 23 2003

Keywords

• nonn

Comments

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).

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

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
• F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
• F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
• F. A. Haight, Letter to N. J. A. Sloane, n.d.

Crossrefs

Cf. A089944, A089945.

A diagonal of A259334.

Programs

Magma

[2*(n+1) * (n+2)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 14 2017

Mathematica

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 *)
Table[2*(n+1)*(n+2)^(n-1), {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

PARI

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

Formula

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

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

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