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

0,5

Comments

The sum of row r of the triangle is (-1)^r (see A244119). - Stanislav Sykora, Jun 21 2014

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

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)).

Example

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; ...

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

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 *)

Prog

(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)}

Crossrefs

Cf. A161552, A244119, A273954.

Sequence in context: A137663 A370983 A257740 * A244119 A122059 A324906

Adjacent sequences: A161625 A161626 A161627 * A161629 A161630 A161631

Keyword

sign,tabl

Author

Paul D. Hanna, Jun 15 2009, Jun 16 2009, Jun 17 2009

Status

approved