A178949 E.g.f. satisfies: A(x) = exp(x^2*A(x)) where A(x) = Sum_{n>=0} a(n)*x^(2n)/(2n)!.

1, 2, 36, 1920, 210000, 39191040, 11181360960, 4534378168320, 2481970620729600, 1764322560000000000, 1580868516481859404800, 1743505552795995891302400, 2321376488366363008816435200, 3671767205084150828189614080000

Offset

0,2

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..100

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

a(n) = (n+1)^(n-1)*(2*n)!/n!.

E.g.f.: LambertW(-x^2)/(-x^2) = Sum_{n>=0} a(n)*x^(2n)/(2n)!.

Example

E.g.f.: A(x) = 1 + 2*x^2/2! + 36*x^4/4! + 1920*x^6/6! +...
log(A(x)) = x^2 + 2*x^4/2! + 36*x^6/4! + 1920*x^8/6! +...

Mathematica

Table[(n+1)^(n-1)(2n)!/n!, {n, 0, 15}] (* Harvey P. Dale, Oct 21 2011 *)

Prog

(PARI) {a(n)=(n+1)^(n-1)*(2*n)!/n!}
(PARI) N=50; /* up to order N */
A(x)=sum(n=0, N-1, if (n%2==1, 0, (n/2+1)^(n/2-1)/(n/2)!*x^n) )+O(x^N); /* e.g.f. */
v=Vec(serlaplace(A(x))) /* gives sequence as vector with interpolated zeros */
/* Now check that e.g.f. satisfies functional equation: */
A(x)-exp(x^2*A(x)) /* ==O(x^50) "==zero" */
(PARI)
N = 28; x = 'x + O('x^N); y = 'y; Fxy = exp(x^2*y) - y;
seq() = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
select(x->x, Vec(serlaplace(Ser(seq()))))  \\ Gheorghe Coserea, Nov 30 2016

Crossrefs

Sequence in context: A174881 A126934 A303503 * A200571 A213985 A203021

Adjacent sequences: A178946 A178947 A178948 * A178950 A178951 A178952

Keyword

nonn

Author

Vladimir Kruchinin, Dec 31 2010

Extensions

Edited by Paul D. Hanna, Jan 03 2011

Status

approved