A340330 G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) satisfies: 1 = Sum_{n>=0} x^n/(1 + 2^n*x*A(x)^n).

1, 1, -23, 3393, -1917839, 4084439425, -33891092958375, 1115079034134766465, -146344823494402837552287, 76757909901662702029167652609, -160993058320324275979629230976271287, 1350558570374627603824657269225692538358977, -45317735199913991775812306272864857418386056626479

Offset

0,3

Comments

Note that the g.f. is an even function, so only the coefficients of even powers of x are shown.

The g.f. A(x) 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, and r = 2*A(x), where A(x) causes the sum to equal unity.

Links

Table of n, a(n) for n=0..12.

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) satisfies the following relations.

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

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

(3) 1 = Sum_{n>=0} (-1)^n * x^(2*n) * 2^(n^2) * A(x)^(n^2) * (1 + x^2*4^n*A(x)^(2*n)) / (1 - x^2*4^n*A(x)^(2*n)).

Example

G.f.: A(x) = 1 + x^2 - 23*x^4 + 3393*x^6 - 1917839*x^8 + 4084439425*x^10 - 33891092958375*x^12 + 1115079034134766465*x^14 - 146344823494402837552287*x^16 + ...
where
1 = 1/(1 + x) + x/(1 + 2*x*A(x)) + x^2/(1 + 2^2*x*A(x)^2) + x^3/(1 + 2^3*x*A(x)^3) + x^4/(1 + 2^4*x*A(x)^4) + x^5/(1 + 2^5*x*A(x)^5) + ...
also
1 = 1/(1 - x) - x/(1 - 2*x*A(x)) + x^2/(1 - 2^2*x*A(x)^2) - x^3/(1 - 2^3*x*A(x)^3) + x^4/(1 - 2^4*x*A(x)^4) - x^5/(1 - 2^5*x*A(x)^5) + ...

Prog

(PARI) {a(n) = my(A=1, V=[1]); for(i=0, 2*n, V=concat(V, 0); A = Ser(concat(V, 0)); V[#V] = polcoeff( sum(m=0, #V+1, x^m/(1 + 2^m*x*A^m) ), #V+1)/2 ); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A340331, A340356.

Sequence in context: A152521 A267692 A136363 * A290786 A248335 A222031

Adjacent sequences: A340327 A340328 A340329 * A340331 A340332 A340333

Keyword

sign

Author

Paul D. Hanna, Jan 08 2021

Status

approved