A340891 G.f. A(x) satisfies: A(x) = (1 - x*A(x)) * Sum_{n>=0} x^n / (1 - x*A(x)^(n+1)).

1, 1, 1, 2, 6, 20, 70, 255, 961, 3726, 14797, 59986, 247606, 1038632, 4420837, 19071954, 83321966, 368400431, 1647706426, 7452622503, 34082926816, 157595263361, 736806253045, 3483636843142, 16660303710511, 80618576499123, 394863246977469, 1958369414771028

Offset

0,4

Comments

The g.f. of this sequence is motivated by the following identity:

Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;

here, p = x, q = x*A(x), and r = A(x).

Links

Table of n, a(n) for n=0..27.

Formula

G.f. A(x) satisfies:

(1) A(x) = (1 - x*A(x)) * Sum_{n>=0} x^n / (1 - x*A(x)^(n+1)).

(2) A(x) = (1 - x*A(x)) * Sum_{n>=0} x^n*A(x)^n / (1 - x*A(x)^n).

(3) A(x) = (1 - x*A(x)) * Sum_{n>=0} x^(2*n) * A(x)^(n^2+n) * (1 - x^2*A(x)^(2*n+1)) / ((1 - x*A(x)^(n+1))*(1 - x*A(x)^n)). - Paul D. Hanna, Feb 20 2021

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 255*x^7 + 961*x^8 + 3726*x^9 + 14797*x^10 + 59986*x^11 + 247606*x^12 + ...
where
A(x)/(1 - x*A(x))  =  1/(1 - x*A(x)) + x/(1 - x*A(x)^2) + x^2/(1 - x*A(x)^3) + x^3/(1 - x*A(x)^4) + x^4/(1 - x*A(x)^5) + ...
also
A(x)/(1 - x*A(x))  =  1/(1-x) + x*A(x)/(1 - x*A(x)) + x^2*A(x)^2/(1 - x*A(x)^2) + x^3*A(x)^3/(1 - x*A(x)^3) + x^4*A(x)^4/(1 - x*A(x)^4) + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = (1 - x*A) * sum(m=0, n, x^m / (1 - x*A^(m+1) +x*O(x^n)) ) ); polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = (1 - x*A) * sum(m=0, n, x^m*A^m / (1 - x*A^m +x*O(x^n)) ) ); polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A340361.

Sequence in context: A192540 A369630 A185202 * A049140 A372526 A092413

Adjacent sequences: A340888 A340889 A340890 * A340892 A340893 A340894

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2021

Status

approved