A161552 E.g.f. satisfies: A(x,y) = exp(x*y*exp(x*A(x,y))).

1, 0, 1, 0, 2, 1, 0, 3, 12, 1, 0, 4, 72, 48, 1, 0, 5, 320, 810, 160, 1, 0, 6, 1200, 8640, 6480, 480, 1, 0, 7, 4032, 70875, 143360, 42525, 1344, 1, 0, 8, 12544, 489888, 2240000, 1792000, 244944, 3584, 1, 0, 9, 36864, 3000564, 27869184, 49218750, 18579456, 1285956, 9216, 1

Offset: 0

Author

Paul D. Hanna, Jun 13 2009, Jun 14 2009

Keywords

• nonn
• tabl

Comments

E.g.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n!.

Row sums, (n+1)^(n-1), equal A000272 (number of trees on n labeled nodes).

Examples

			Triangle begins:
1;
0,1;
0,2,1;
0,3,12,1;
0,4,72,48,1;
0,5,320,810,160,1;
0,6,1200,8640,6480,480,1;
0,7,4032,70875,143360,42525,1344,1;
0,8,12544,489888,2240000,1792000,244944,3584,1;
0,9,36864,3000564,27869184,49218750,18579456,1285956,9216,1; ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A000272, A072590; A161565, A161566, A161567, A141369.

Programs

Mathematica

Join[{1}, Table[Binomial[n, k]*(n - k + 1)^(k - 1)*k^(n - k), {n, 1, 10}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 18 2017 *)

PARI

{T(n,k)=binomial(n,k)*(n-k+1)^(k-1)*k^(n-k)}

PARI

{T(n,k)=local(A=1+x); for(i=0,n, A=exp(x*y*exp(x*A+O(x^n)))); n!*polcoeff(polcoeff(A,n,x),k,y)}

Formula

T(n,k) = binomial(n,k) * (n-k+1)^(k-1) * k^(n-k).

E.g.f. A(x,y) at y=1: A(x,1) = LambertW(-x)/(-x).

From Paul D. Hanna, Jun 14 2009: (Start)

More generally, if G(x) = exp(p*x*exp(q*x*G(x))),

where G(x)^m = Sum_{n>=0} g(n,m)*x^n/n!,

then g(n,m) = Sum_{k=0..n} C(n,k)*p^k*q^(n-k)*m*(n-k+m)^(k-1)*k^(n-k).

(End)