0,5

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).

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).

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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)

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; ...

(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)}

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

Sequence in context: A153007 A090683 A117652 this_sequence A095859 A088850 A143424

Adjacent sequences: A161549 A161550 A161551 this_sequence A161553 A161554 A161555

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 13 2009, Jun 14 2009