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

0,2

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.

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)).

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 *)

Prog

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

Crossrefs

Cf. A089944, A089945.

A diagonal of A259334.

Sequence in context: A138419 A366341 A341727 * A343094 A012244 A342168

Adjacent sequences: A089943 A089944 A089945 * A089947 A089948 A089949

Keyword

nonn

Author

Paul D. Hanna, Nov 23 2003

Status

approved