_Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 10 2010

Continued fraction formula corrected by _Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 13 2010

reviewed

approved

proposed

reviewed

A(A(x)) = x*Sum_{n>=01} A(x)^n*Product_{k=1..n} (1-x*q^(4*k-3))/(1-x*q^(4*k-1)) due to a q-series identity.

A(A(x)) = x-1 + 1/(1- q*x/(1- (q^3-q)*x/(1- q^5*x/(1- (q^7-q^3)*x/(1- q^9*x/(1- (q^11-q^5)*x/(1- q^13*x/(1- (q^15-q^7)*x/(1- ...)))))))))

Continued fraction formula corrected by Paul D. Hanna (pauldhanna(AT)juno.com), Dec 13 2010

reviewed

proposed

proposed

reviewed

G.f.: A(A(x)) = x*Sum_{n>=0} A(x)^n*Product_{k=1..n} (1-x*q^(4*k-3))/(1-x*q^(4*k-1)) due to a q-series identity.

G.f. satisfies: A(A(x)) = Sum_{n>=1} A(x)^(n^2)/x^(n^2-n).

(1) G.f.: A(x) = Sum_{n>=1} x^(n^2)/G(x)^(n^2-n) where G(x) is the series reversion of A(x).

A(A(x)) = 1x/(1- q*x/(1- (q^3-q)*x/(1- q^5*x/(1- (q^7-q^3)*x/(1- q^9*x/(1- (q^11-q^5)*x/(1- q^13*x/(1- (q^15-q^7)*x/(1- ...)))))))))

G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 261*x^6 + 1344*x^7 +...

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

allocated for Paul D. Hanna

G.f. satisfies: A(A(x)) = Sum_{n>=1} A(x)^(n^2)/x^(n^2-n).

1, 1, 3, 12, 54, 261, 1344, 7380, 43099, 265739, 1717466, 11586670, 81422063, 594828360, 4508090145, 35380563603, 287130931064, 2406309163514, 20797551211656, 185158224231178, 1696132889163096, 15969702544475270

1,3

(1) G.f.: A(x) = Sum_{n>=1} x^(n^2)/G(x)^(n^2-n) where G(x) is the series reversion of A(x).

(2) Let q = A(x)/x, then g.f. A(x) satisfies the continued fraction:

A(A(x)) = 1/(1- q*x/(1- (q^3-q)*x/(1- q^5*x/(1- (q^7-q^3)*x/(1- q^9*x/(1- (q^11-q^5)*x/(1- q^13*x/(1- (q^15-q^7)*x/(1- ...)))))))))

due to an identity of a partial elliptic theta function.

(3) Let q = A(x)/x, then g.f. A(x) satisfies:

G.f.: A(A(x)) = x*Sum_{n>=0} A(x)^n*Product_{k=1..n} (1-x*q^(4*k-3))/(1-x*q^(4*k-1)) due to a q-series identity.

G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 261*x^6 + 1344*x^7 +...

where:

A(A(x)) = A(x) + A(x)^4/x^2 + A(x)^9/x^6 + A(x)^16/x^12 + A(x)^25/x^20 +...

Explicitly,

A(A(x)) = x + 2*x^2 + 8*x^3 + 40*x^4 + 222*x^5 + 1314*x^6 + 8172*x^7 + 53049*x^8 + 357905*x^9 + 2500608*x^10 +...

Related expansions:

A(x)^4/x^2 = x^2 + 4*x^3 + 18*x^4 + 88*x^5 + 451*x^6 + 2388*x^7 +...

A(x)^9/x^6 = x^3 + 9*x^4 + 63*x^5 + 408*x^6 + 2556*x^7 +...

A(x)^16/x^12 = x^4 + 16*x^5 + 168*x^6 + 1472*x^7 + 11684*x^8 +...

A(x)^25/x^20 = x^5 + 25*x^6 + 375*x^7 + 4400*x^8 + 44600*x^9 +...

A(x)^36/x^30 = x^6 + 36*x^7 + 738*x^8 + 11352*x^9 + 145899*x^10 +...

...

Let G(x) satisfy A(G(x)) = x, then

A(x) = x + x^4/G(x)^2 + x^9/G(x)^6 + x^16/G(x)^12 + x^25/G(x)^20 +...

where:

G(x) = x - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 42*x^7 - 303*x^8 - 2000*x^9 - 11804*x^10 - 70275*x^11 - 459489*x^12 +...

Related expansions:

x^4/G(x)^2 = x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 40*x^6 + 116*x^7 +...

x^9/G(x)^6 = x^3 + 6*x^4 + 27*x^5 + 110*x^6 + 423*x^7 + 1566*x^8 +...

x^16/G(x)^12 = x^4 + 12*x^5 + 90*x^6 + 544*x^7 + 2895*x^8 +...

x^25/G(x)^20 = x^5 + 20*x^6 + 230*x^7 + 2000*x^8 + 14605*x^9 +...

x^36/G(x)^30 = x^6 + 30*x^7 + 495*x^8 + 5950*x^9 + 58245*x^10 +...

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

allocated

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 10 2010

approved

proposed

allocated for Paul D. Hanna

recycled

allocated