A202135 G.f. satisfies: A(x) = 1 + (eta(x^2*A(x)^2)^10 / (eta(x*A(x))^4 * eta(x^4*A(x)^4)^4) - 1)/4, where eta(q) is the Dedekind eta function without the q^(1/24) factor.

1, 1, 2, 4, 10, 28, 84, 260, 818, 2609, 8434, 27622, 91524, 306270, 1033364, 3510960, 12000898, 41239249, 142387668, 493732424, 1718653462, 6003501630, 21037928794, 73937288814, 260544327156, 920376228329, 3258619058158, 11561493797242, 41100002807572

Offset

0,3

Links

Table of n, a(n) for n=0..28.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) A(x) = 1 + Sum_{n>=1} (-1)^[n/2] * (x*A(x))^(n*(n+1)/2) / (1 + (-x*A(x))^n);

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

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

Let G(x) = 1 + g(x) where g(x) is the g.f. of A002654, then:

(4) A(x) = G(x*A(x)) where G(x) = A(x/G(x));

(5) A(x) = (1/x)*Series_Reversion( x/G(x));

(6) a(n) = [x^n] G(x)^(n+1) / (n+1);

where A002654(n) is the number of ways of writing n as a sum of at most two nonzero squares (with respect to order).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 84*x^6 +...
where the g.f. A = A(x) satisfies the series identities:
(1) A(x) = 1 + x*A/(1 - x*A) - x^3*A^3/(1 + x^2*A^2) - x^6*A^6/(1 - x^3*A^3) + x^10*A^10/(1 + x^4*A^4) + x^15*A^15/(1 - x^5*A^5) +...
(2) A(x) = 1 + x*A/(1 + x^2*A^2) + x^2*A^2/(1 + x^4*A^4) + x^3*A^3/(1 + x^6*A^6) + x^4*A^4/(1 + x^8*A^8) +...
(3) A(x) = 1 + x*A/(1 - x*A) - x^3*A^3/(1 - x^3*A^3) + x^5*A^5/(1 - x^5*A^5) - x^7*A^7/(1 - x^7*A^7) +...
(4) G.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = 1 + g.f. of A002654:
A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^4*A(x)^4 + 2*x^5*A(x)^5 + x^8*A(x)^8 + x^9*A(x)^9 + 2*x^10*A(x)^10 +...+ A002654(n)*x^n*A(x)^n +...
(6) Coefficients in the powers of G(x) = 1 + g.f. of A002654 begin:
G^1: [(1), 1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 1, 2, ...];
G^2: [1,(2), 3, 2, 3, 6, 6, 4, 3, 8, 12, 6, 6, 10, 12, 12, 3, ...];
G^3: [1, 3, (6), 7, 9, 15, 22, 24, 21, 30, 48, 51, 46, 51, ...];
G^4: [1, 4, 10,(16), 23, 36, 58, 80, 91, 112, 162, 216, ...];
G^5: [1, 5, 15, 30,(50), 81, 135, 210, 285, 370, 511, 720, ...];
G^6: [1, 6, 21, 50, 96, (168), 291, 486, 741, 1052, ...];
G^7: [1, 7, 28, 77, 168, 322, (588), 1037, 1715, 2646, ...];
G^8: [1, 8, 36, 112, 274, 576, 1120,(2080), 3663, 6064, ...];
G^9: [1, 9, 45, 156, 423, 972, 2022, 3960, (7362), 12940, ...]; ...
where the coefficients in parenthesis form the initial terms of this sequence:
A = [1/1, 2/2, 6/3, 16/4, 50/5, 168/6, 588/7, 2080/8, 7362/9, ...].

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+(eta(x^2*A^2)^10/(eta(x*A)^4*eta(x^4*A^4)^4)-1)/4+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, (-1)^(m\2)*(x*A)^(m*(m+1)/2)/(1+(-x*A+x*O(x^n))^m))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, x^m*(A+x*O(x^n))^m/(1+x^(2*m)*(A+x*O(x^n))^(2*m)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, (-1)^(m-1)*x^(2*m-1)*(A+x*O(x^n))^(2*m-1)/(1-x^(2*m-1)*(A+x*O(x^n))^(2*m-1)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x, G=1+sum(m=1, n\2+1, (-1)^(m-1)*x^(2*m-1)/(1-x^(2*m-1)+x^2*O(x^n)))); A=1/x*serreverse(x/G); polcoeff(A, n)}

Crossrefs

Cf. A002654.

Sequence in context: A149824 A331938 A302146 * A068875 A262543 A348197

Adjacent sequences: A202132 A202133 A202134 * A202136 A202137 A202138

Keyword

nonn

Author

Paul D. Hanna, Dec 11 2011

Status

approved