|
| |
|
|
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.
|
|
0
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
LINKS
|
Table of n, a(n) for n=0..44.
|
|
|
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; ...
|
|
|
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.
Sequence in context: A194365 A216217 A137663 * A122059 A164917 A166238
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
|
| |
|
|
