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%I #12 Nov 18 2017 04:26:13
%S 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,
%T 6480,480,1,0,7,4032,70875,143360,42525,1344,1,0,8,12544,489888,
%U 2240000,1792000,244944,3584,1,0,9,36864,3000564,27869184,49218750,18579456,1285956,9216,1
%N E.g.f. satisfies: A(x,y) = exp(x*y*exp(x*A(x,y))).
%C E.g.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n!.
%C Row sums, (n+1)^(n-1), equal A000272 (number of trees on n labeled nodes).
%H G. C. Greubel, <a href="/A161552/b161552.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F T(n,k) = binomial(n,k) * (n-k+1)^(k-1) * k^(n-k).
%F E.g.f. A(x,y) at y=1: A(x,1) = LambertW(-x)/(-x).
%F From _Paul D. Hanna_, Jun 14 2009: (Start)
%F More generally, if G(x) = exp(p*x*exp(q*x*G(x))),
%F where G(x)^m = Sum_{n>=0} g(n,m)*x^n/n!,
%F 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).
%F (End)
%e Triangle begins:
%e 1;
%e 0,1;
%e 0,2,1;
%e 0,3,12,1;
%e 0,4,72,48,1;
%e 0,5,320,810,160,1;
%e 0,6,1200,8640,6480,480,1;
%e 0,7,4032,70875,143360,42525,1344,1;
%e 0,8,12544,489888,2240000,1792000,244944,3584,1;
%e 0,9,36864,3000564,27869184,49218750,18579456,1285956,9216,1; ...
%t 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 *)
%o (PARI) {T(n,k)=binomial(n,k)*(n-k+1)^(k-1)*k^(n-k)}
%o (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)}
%Y Cf. A000272, A072590; A161565, A161566, A161567, A141369.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Jun 13 2009, Jun 14 2009
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