A161628 - OEIS

%I #14 Jan 29 2023 10:53:30

%S 1,0,-1,0,-2,3,0,-3,18,-16,0,-4,72,-192,125,0,-5,240,-1440,2500,-1296,

%T 0,-6,720,-8640,30000,-38880,16807,0,-7,2016,-45360,280000,-680400,

%U 705894,-262144,0,-8,5376,-217728,2240000,-9072000,16941456,-14680064,4782969

%N 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.

%C The sum of row r of the triangle is (-1)^r (see A244119). - _Stanislav Sykora_, Jun 21 2014

%H G. C. Greubel, <a href="/A161628/b161628.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F T(n,k) = (-1)^k*C(n,k)*(k+1)^(k-1)*k^(n-k).

%F E.g.f. satisfies: A(x,y) = exp(-x*y*exp(x)*A(x,y)).

%F E.g.f.: A(x,y) = Sum_{n>=0} (n+1)^(n-1) * (-x)^n*y^n*exp(n*x)/n!.

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

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

%F then g(n,m) = C(n,k)*p^k*q^(n-k) * m*(k+m)^(k-1) * k^(n-k)

%F and G(x,y) = LambertW(-p*x*y*exp(q*x))/(-p*x*y*exp(q*x)).

%e Triangle begins:

%e 1;

%e 0, -1;

%e 0, -2, 3;

%e 0, -3, 18, -16;

%e 0, -4, 72, -192, 125;

%e 0, -5, 240, -1440, 2500, -1296;

%e 0, -6, 720, -8640, 30000, -38880, 16807;

%e 0, -7, 2016, -45360, 280000, -680400, 705894, -262144;

%e 0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969;

%e 0, -9, 13824, -979776, 16128000, -102060000, 304946208, -462422016, 344373768, -100000000; ...

%p A161628 := (n, k) -> (-1)^k*binomial(n, k)*(k+1)^(k-1)*k^(n-k):

%p seq(seq(A161628(n,k), k=0..n), n=0..8); # _Peter Luschny_, Jan 29 2023

%t Join[{1}, Table[(-1)^k*Binomial[n, k]*(k + 1)^(k - 1)*k^(n - k), {n, 1, 10}, {k, 0, n}]] // Flatten (* _G. C. Greubel_, Nov 09 2017 *)

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

%o (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 (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 Cf. A161552, A244119, A273954.

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, Jun 15 2009, Jun 16 2009, Jun 17 2009