OFFSET
0,2
FORMULA
The g.f. A(x) allows the following series to be equal:
(1) B(x) = Sum_{n>=0} A(x)^(n^2) * x^n.
(2) B(x) = Sum_{n>=0} ((1+x)^n - 1)^n.
(3) B(x) = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 3*x + 18*x^2 + 184*x^3 + 2742*x^4 + 51650*x^5 + 1148054*x^6 + 29089167*x^7 + 823981958*x^8 + 25773170170*x^9 + 882457387327*x^10 + ...
such that the following sum
B(x) = 1 + A(x)*x + A(x)^4*x^2 + A(x)^9*x^3 + A(x)^16*x^4 + A(x)^25*x^5 + A(x)^36*x^6 + A(x)^49*x^7 + A(x)^64*x^8 + ... + A(x)^(n^2)*x^n + ...
equals
B(x) = 1 + ((1+x) - 1) + ((1+x)^2 - 1)^2 + ((1+x)^3 - 1)^3 + ((1+x)^4 - 1)^4 + ((1+x)^5 - 1)^5 + ((1+x)^6 - 1)^6 + ... + ((1+x)^n - 1)^n + ...
as well as
B(x) = 1/2 + (1+x)/(1 + (1+x))^2 + (1+x)^4/(1 + (1+x)^2)^3 + (1+x)^9/(1 + (1+x)^3)^4 + (1+x)^16/(1 + (1+x)^4)^5 + ... + (1+x)^(n^2)/(1 + (1+x)^n)^(n+1) + ...
where
B(x) = 1 + x + 4*x^2 + 31*x^3 + 338*x^4 + 4769*x^5 + 82467*x^6 + 1687989*x^7 + 39905269*x^8 + 1069863695*x^9 + ... + A122400(n)*x^n + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, ((1+x)^m - 1 +x*O(x^#A))^m - x^m*Ser(A)^(m^2) ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 28 2019
STATUS
approved