A088713 G.f. A(x) satisfies A(x/A(x)) = 1/(1-x).

1, 1, 2, 6, 24, 118, 674, 4308, 30062, 225266, 1791964, 15009118, 131566314, 1201452248, 11389283418, 111761444078, 1132680800640, 11834071103246, 127261591139010, 1406778021294220, 15967144849210158, 185897394076705298

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

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

G.f.: A(x*g(x)) = g(x) = (1-1/A(x))/x where g(x) is the g.f. of A088714.

From Paul D. Hanna, Dec 06 2009: (Start)

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

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

(End)

Logarithmic derivative: given g.f. A(x), let G(x) = A(x*G(x)) be the g.f. of A088714, then A'(x)/A(x) = (G(x) + x*G'(x)) / (1 - x*G(x)).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 118*x^5 + 674*x^6 +...
Illustration of logarithmic derivation.
If we form an array of coefficients of x^k in A(x)^n, n>=1, like so:
A^1: [1],1,  2,   6,   24,   118,   674,    4308, ...;
A^2: [1, 2], 5,  16,   64,   308,  1716,   10724, ...;
A^3: [1, 3,  9], 31,  126,   600,  3278,   20070, ...;
A^4: [1, 4, 14,  52], 217,  1032,  5560,   33440, ...;
A^5: [1, 5, 20,  80,  345], 1651,  8820,   52270, ...;
A^6: [1, 6, 27, 116,  519,  2514],13385,   78420, ...;
A^7: [1, 7, 35, 161,  749,  3689, 19663], 114269, ...; ...
then the sums of the coefficients of x^k, k=0..n-1, in A(x)^n (shown above in brackets) begin:
1 = 1;
1 + 2 = 3;
1 + 3 +  9 = 13;
1 + 4 + 14 +  52 = 71;
1 + 5 + 20 +  80 +  345 = 451;
1 + 6 + 27 + 116 +  519 +  2514 = 3183;
1 + 7 + 35 + 161 +  749 +  3689 + 19663 = 24305; ...
and equal the coefficients in log(A(x)):
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 71*x^4/4 + 451*x^5/5 + 3183*x^6/6 + 24305*x^7/7 + 197551*x^8/8 +...
The main diagonal in the above table forms the g.f. G(x) of A088714:
[1/1, 2/2, 9/3, 52/4, 345/5, 2514/6, 19663/7, ...]
where G(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...
satisfies A'(x)/A(x) = (G(x) + x*G'(x)) / (1 - x*G(x)).

Mathematica

terms = 22; A[_] = 1; Do[A[x_] = 1 + x*A[x]*A[1 - 1/A[x]] + O[x]^j // Normal, {j, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)

Prog

(PARI) a(n)=local(A=1+x); for(i=1, n, A=(1+A*serreverse(x/(A+x*O(x^n))))^1); polcoeff(A, n)
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 06 2009
(PARI) {a(n)=local(A=1+x); if(n==0, 1, for(i=1, n,
A=1+x*exp(sum(k=1, n-1, sum(j=0, k, polcoeff(A^k+x*O(x^j), j))*x^k/k)+x*O(x^n))));
polcoeff(A+x*O(x^n), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 09 2013

Crossrefs

Cf. A088714.

Variants: A154677, A168448, A168449, A168478, A168479.

Sequence in context: A247472 A224295 A263777 * A193938 A298432 A336072

Adjacent sequences: A088710 A088711 A088712 * A088714 A088715 A088716

Keyword

nonn

Author

Paul D. Hanna, Oct 12 2003

Status

approved