OFFSET
0,4
COMMENTS
Compare to the curious identity: Sum_{n=-oo..+oo} x^n/a^(n^2+n) * (a^n - x^n)^n = 0 for all fixed a.
FORMULA
G.f. satisfies: A(x) = 1+x + x*Sum_{n=-oo..+oo} (x*A(x))^(n^2-n) / (x^n - A(x)^n)^n.
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 11*x^4 + 54*x^5 + 224*x^6 + 1221*x^7 +...
such that A(x) = 1 + 2*x + x*N(x) + x*P(x) where
N(x) = Sum_{n<=-1} x^n*A(x)^n * (A(x)^n - x^n)^n,
P(x) = Sum_{n>=1} x^n*A(x)^n * (A(x)^n - x^n)^n;
Equivalently,
N(x) = Sum_{n>=1} (x*A(x))^(n^2-n) / (x^n - A(x)^n)^n,
P(x) = Sum_{n<=-1} (x*A(x))^(n^2-n) / (x^n - A(x)^n)^n,
Explicitly,
N(x) = -1 + 2*x^2 + 2*x^3 + 13*x^4 + 32*x^5 + 199*x^6 + 926*x^7 + 5518*x^8 + 31303*x^9 + 194051*x^10 + 1226544*x^11 + 8123893*x^12 + 55526652*x^13 + 393267242*x^14 + 2876241417*x^15 + 21725489552*x^16 +...
P(x) = x + 2*x^2 + 9*x^3 + 41*x^4 + 192*x^5 + 1022*x^6 + 5580*x^7 + 32464*x^8 + 196550*x^9 + 1242104*x^10 + 8143613*x^11 + 55354509*x^12 + 389330167*x^13 + 2831082490*x^14 + 21265861682*x^15 + 164909939950*x^16 +...
PROG
(PARI) {a(n) = local(A=1+x+x*O(x^n));
for(i=1, n, A=1+x + x*sum(k=-n-1, n+1, x^k*A^k * (A^k-x^k)^k)); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
(PARI) {a(n) = local(A=1+x+x*O(x^n));
for(i=1, n, A=1+x + x*sum(k=-n-1, n+1, (x*A)^(k^2-k) / (x^k-A^k)^k)); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 21 2015
STATUS
approved