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A161552
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E.g.f. satisfies: A(x,y) = exp(x*y*exp(x*A(x,y))).
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2
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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).
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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).
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)
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EXAMPLE
| 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; ...
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PROG
| (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)}
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CROSSREFS
| Cf. A000272, A072590; A161565, A161566, A161567, A141369.
Sequence in context: A077874 A153007 A090683 * A095859 A191897 A088850
Adjacent sequences: A161549 A161550 A161551 * A161553 A161554 A161555
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 13 2009, Jun 14 2009
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