A213591 G.f. A(x) satisfies: A( x - A(x)^2 ) = x.

1, 1, 4, 24, 178, 1512, 14152, 142705, 1528212, 17211564, 202460400, 2474708496, 31310415376, 408815254832, 5495451727376, 75907303147652, 1075685334980240, 15618612118252960, 232102241507321384, 3526880759915999016, 54755450619399484512, 867928449982022915984

Offset

1,3

Comments

Unsigned version of A139702.

Self-convolution is A276370.

Row sums of triangle A277295.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Formula

G.f. satisfies:

(1) A(x) = x + A(A(x))^2.

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.

(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).

(4) A(x) = x*G(A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.

(5) A(x)^2 = x*F(A(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628.

(6) x = A(A( x-x^2 - A(x)^2 )). - Paul D. Hanna, Jul 01 2012

(7) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:

A = x + B^2;

B = A + C^2;

C = B + D^2;

D = C + E^2; ...

where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc.

...

a(n) = Sum_{k=0..n-1} A277295(n,k).

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
where A(x) = x + A(A(x))^2:
A(A(x)) = x + 2*x^2 + 10*x^3 + 69*x^4 + 568*x^5 + 5250*x^6 + 52792*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
The g.f. satisfies the series:
A(x) = x + A(x)^2 + d/dx A(x)^4/2! + d^2/dx^2 A(x)^6/3! + d^3/dx^3 A(x)^8/4! +...
Logarithmic series:
log(A(x)/x) = A(x)^2/x + [d/dx A(x)^4/x]/2! + [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! +...
Also, A(x) = x*G(A(x)^2/x) where G(x) = x/A(x/G(x)^2) is the g.f. of A212411:
G(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 + 15261*x^7 +...
Also, A(x)^2 = x*F(A(x)) where F(x) is the g.f. of A213628:
F(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 + 46013*x^8 +...

Mathematica

terms = 22; A[_] = 0; Do[A[x_] = x + A[A[x]]^2 + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 09 2018 *)

Prog

(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^2+x*O(x^n))); polcoeff(A, n))}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 21, print1(a(n), ", "))

Crossrefs

Cf. A220379, A139702, A212411, A138740, A213628, A213639.

Cf. A088714, A088717, A091713, A120971, A140094, A140095.

Cf. A143426, A087949, A143435, A182969.

Cf. A275765, A276360, A276361, A276362, A276363, A276370.

Cf. A277295 (triangle).

Sequence in context: A007846 A337507 A139702 * A243689 A309637 A168452

Adjacent sequences: A213588 A213589 A213590 * A213592 A213593 A213594

Keyword

nonn

Author

Paul D. Hanna, Jun 14 2012

Status

approved