OFFSET
1,3
FORMULA
(1) G.f.: A(x) = Sum_{n>=1} x^(n(n+1)/2)/G(x)^(n(n-1)/2) where G(x) is the series reversion of A(x).
(2) Let q = A(x)/x then g.f. A(x) satisfies:
A(A(x)) = Sum_{n>=1} A(x)^n*Product_{k=1..n} (1-x*q^(2k-1))/(1-x*q^(2k))
due to a q-series identity.
(3) Let q = A(x)/x, then g.f. A(x) satisfies the continued fraction:
A(A(x)) = -1 + 1/(1- q*x/(1- (q^2-q)*x/(1- q^3*x/(1- (q^4-q^2)*x/(1- q^5*x/(1- (q^6-q^3)*x/(1- q^7*x/(1- (q^8-q^4)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 43*x^6 + 144*x^7 +...
where:
A(A(x)) = A(x) + A(x)^3/x + A(x)^6/x^3 + A(x)^10/x^6 + A(x)^15/x^10 +...
...
Let q = A(x)/x, then g.f. A(x) satisfies:
A(A(x)) = A(x)*(1-xq)/(1-xq^2) + A(x)^2*(1-xq)(1-xq^3)/((1-xq^2)(1-xq^4)) + A(x)^3*(1-xq)(1-xq^3)(1-xq^5)/((1-xq^2)(1-xq^4)(1-xq^6)) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 324*x^6 + 1380*x^7 + 6137*x^8 + 28348*x^9 + 135549*x^10 + 669406*x^11 + 3408490*x^12 +...
Related expansions are:
A(x)^3/x = x^2 + 3*x^3 + 9*x^4 + 28*x^5+ 90*x^6 + 300*x^7 +...
A(x)^6/x^3 = x^3 + 6*x^4 + 27*x^5 + 110*x^6 + 429*x^7 +...
A(x)^10/x^6 = x^4 + 10*x^5 + 65*x^6 + 350*x^7 + 1700*x^8 +...
A(x)^15/x^10 = x^5 + 15*x^6 + 135*x^7 + 950*x^8 + 5775*x^9 +...
A(x)^21/x^15 = x^6 + 21*x^7 + 252*x^8 + 2275*x^9 + 17199*x^10 +...
...
Let G(x) satisfy A(G(x)) = x, then
A(x) = x + x^3/G(x) + x^6/G(x)^3 + x^10/G(x)^6 + x^15/G(x)^10 +...
where:
G(x) = x - x^2 - x^6 - 4*x^7 - 9*x^8 - 26*x^9 - 129*x^10 - 537*x^11 - 1961*x^12 - 9088*x^13 - 44722*x^14 - 199057*x^15 -...
PROG
(PARI) {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=sum(m=1, n, x^(m*(m+1)/2)/serreverse(A)^(m*(m-1)/2))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 12 2010
EXTENSIONS
Made cosmetic change to example. All formulas have been verified.
STATUS
approved