A177384 G.f. satisfies: A(x) = 1 + x/(A(x) + x*A'(x)).

1, 1, -2, 10, -72, 644, -6704, 78408, -1008480, 14065744, -210682080, 3365194560, -57019105920, 1020662366400, -19238635678208, 380825404556288, -7898501807543808, 171304216873595136, -3878189367387230720

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Formula

G.f. satisfies: d/dx x*A(x) = -x + (1/x)*d/dx { [x*A(x)]^2/2 }.

Let F(x) be the g.f. of A177383, then

. a(n) = [x^n] F(x)^(n+1)/(n+1);

. A(x) = F(x*A(x)) so that A(x) = (1/x)*Series_Reversion(x/F(x))

where F(x) satisfies: [x^n] F(x)^(n+1) = [x^n] F(x)^(n+2).

a(n) ~ (-1)^(n+1) * c * n! * n^4, where c = 0.005428317993266202636748034138.... - Vaclav Kotesovec, Feb 22 2014

Recurrence: a(0) = a(1) = 1, a(n) = -Sum_{0<k<n} (k+1)*a(k)*a(n-k). - Vladimir Reshetnikov, Nov 14 2016

A177383(n) / a(n) ~ exp(1). - Vaclav Kotesovec, Mar 06 2020

Example

G.f.: A(x) = 1 + x - 2*x^2 + 10*x^3 - 72*x^4 + 644*x^5 - 6704*x^6 +...
d/dx x*A(x) = 1 + 2*x - 6*x^2 + 40*x^3 - 360*x^4 + 3864*x^5 -...
d/dx [x*A(x)]^2/2 = x + 3*x^2 - 6*x^3 + 40*x^4 - 360*x^5 + 3864*x^6 -...

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = -Sum[(k+1) a[k] a[n-k], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 14 2016 *)

Prog

(PARI) {a(n)=local(G=1+x); for(i=1, n, G=1+x/(G+x*deriv(G)+x*O(x^n))); polcoeff(G, n)}

Crossrefs

Cf. A177383.

Sequence in context: A366241 A321446 A111554 * A354288 A182525 A321389

Adjacent sequences: A177381 A177382 A177383 * A177385 A177386 A177387

Keyword

sign

Author

Paul D. Hanna, May 15 2010

Status

approved