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.
1, 0, -1, 0, -2, 3, 0, -3, 18, -16, 0, -4, 72, -192, 125, 0, -5, 240, -1440, 2500, -1296, 0, -6, 720, -8640, 30000, -38880, 16807, 0, -7, 2016, -45360, 280000, -680400, 705894, -262144, 0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969
Offset: 0
Examples
Triangle begins: 1; 0, -1; 0, -2, 3; 0, -3, 18, -16; 0, -4, 72, -192, 125; 0, -5, 240, -1440, 2500, -1296; 0, -6, 720, -8640, 30000, -38880, 16807; 0, -7, 2016, -45360, 280000, -680400, 705894, -262144; 0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969; 0, -9, 13824, -979776, 16128000, -102060000, 304946208, -462422016, 344373768, -100000000; ...
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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Maple
A161628 := (n, k) -> (-1)^k*binomial(n, k)*(k+1)^(k-1)*k^(n-k): seq(seq(A161628(n,k), k=0..n), n=0..8); # Peter Luschny, Jan 29 2023
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Mathematica
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 *) -
PARI
{T(n,k)=(-1)^k*binomial(n,k)*(k+1)^(k-1)*k^(n-k)} -
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)} -
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)}
Formula
T(n,k) = (-1)^k*C(n,k)*(k+1)^(k-1)*k^(n-k).
E.g.f. satisfies: A(x,y) = exp(-x*y*exp(x)*A(x,y)).
E.g.f.: A(x,y) = Sum_{n>=0} (n+1)^(n-1) * (-x)^n*y^n*exp(n*x)/n!.
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.
More generally, if G(x,y) = exp(p*x*y*exp(q*x)*G(x,y)),
where G(x,y)^m = Sum_{n>=0} g(n,m)*x^n/n!,
then g(n,m) = C(n,k)*p^k*q^(n-k) * m*(k+m)^(k-1) * k^(n-k)
and G(x,y) = LambertW(-p*x*y*exp(q*x))/(-p*x*y*exp(q*x)).
Comments