Search: id:A161628
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%I A161628
%S A161628 1,0,1,0,2,3,0,3,18,16,0,4,72,192,125,0,5,240,1440,2500,1296,0,
%T A161628 6,720,8640,30000,38880,16807,0,7,2016,45360,280000,680400,705894,
%U A161628 262144,0,8,5376,217728,2240000,9072000,16941456,14680064,4782969
%V A161628 1,0,-1,0,-2,3,0,-3,18,-16,0,-4,72,-192,125,0,-5,240,-1440,2500,-1296,
               0,
%W A161628 -6,720,-8640,30000,-38880,16807,0,-7,2016,-45360,280000,-680400,705894,
%X A161628 -262144,0,-8,5376,-217728,2240000,-9072000,16941456,-14680064,4782969
%N A161628 E.g.f.: A(x,y) = LambertW(x*y*exp(x))/(x*y*exp(x)), as a triangle of 
               coefficients T(n,k) = [x^n*y^k/n! ] A(x,y), read by rows.
%F A161628 T(n,k) = (-1)^k*C(n,k)*(k+1)^(k-1)*k^(n-k).
%F A161628 E.g.f. satisfies: A(x,y) = exp(-x*y*exp(x)*A(x,y)).
%F A161628 E.g.f.: A(x,y) = Sum_{n>=0} (n+1)^(n-1) * (-x)^n*y^n*exp(n*x)/n!.
%F A161628 E.g.f.: A(x,y) = (1/x)*Series_Reversion[x*G(x,y)] where G(x,y) = exp(x*y*exp(x*G(x,
               y))) is the e.g.f. of A161552.
%F A161628 More generally, if G(x,y) = exp(p*x*y*exp(q*x)*G(x,y)),
%F A161628 where G(x,y)^m = Sum_{n>=0} g(n,m)*x^n/n!,
%F A161628 then g(n,m) = C(n,k)*p^k*q^(n-k) * m*(k+m)^(k-1) * k^(n-k)
%F A161628 and G(x,y) = LambertW(-p*x*y*exp(q*x))/(-p*x*y*exp(q*x)).
%e A161628 Triangle begins:
%e A161628 1;
%e A161628 0,-1;
%e A161628 0,-2,3;
%e A161628 0,-3,18,-16;
%e A161628 0,-4,72,-192,125;
%e A161628 0,-5,240,-1440,2500,-1296;
%e A161628 0,-6,720,-8640,30000,-38880,16807;
%e A161628 0,-7,2016,-45360,280000,-680400,705894,-262144;
%e A161628 0,-8,5376,-217728,2240000,-9072000,16941456,-14680064,4782969;
%e A161628 0,-9,13824,-979776,16128000,-102060000,304946208,-462422016,344373768,
               -100000000; ...
%o A161628 (PARI) {T(n,k)=(-1)^k*binomial(n,k)*(k+1)^(k-1)*k^(n-k)}
%o A161628 (PARI) {T(n,k)=local(A,LW=serreverse(x*exp(x+x*O(x^n))));A=subst(LW/x,
               x,x*y*exp(x));n!*polcoeff(polcoeff(A,n,x),k,y)}
%o A161628 (PARI) {T(n,k)=local(G=1+x);for(i=0,n,G=exp(x*y*exp(x*G+O(x^n))));n!*polcoeff(polcoeff(serreverse(x*G)/
               x,n,x),k,y)}
%Y A161628 Cf. A161552.
%Y A161628 Sequence in context: A047773 A035549 A137663 this_sequence A122059 A164917 
               A166238
%Y A161628 Adjacent sequences: A161625 A161626 A161627 this_sequence A161629 A161630 
               A161631
%K A161628 sign,tabl
%O A161628 0,5
%A A161628 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 15 2009, Jun 16 2009, Jun 
               17 2009

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