OFFSET
0,5
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..495
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2) / A(x)^n.
(2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)/2) / A(x)^n.
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + 24574*x^12 + 109393*x^13 + 508026*x^14 + ...
such that
A(x) = 1 + (1+x)*x/A(x) + (1+x)^3*x^2/A(x)^2 + (1+x)^6*x^3/A(x)^3 + (1+x)^10*x^4/A(x)^4 + (1+x)^15*x^5/A(x)^5 + (1+x)^21*x^6/A(x)^6 + (1+x)^28*x^7/A(x)^7 + ... + (1+x)^(n*(n+1)/2) * x^n / A(x)^n + ...
Also
1 + x = 1 + x/A(x) + (1+x)*x^2/A(x)^2 + (1+x)^3*x^3/A(x)^3 + (1+x)^6*x^4/A(x)^4 + (1+x)^10*x^5/A(x)^5 + (1+x)^15*x^6/A(x)^6 + (1+x)^21*x^7/A(x)^7 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, (1+x +x*O(x^#A))^(n*(n+1)/2) * x^n/Ser(A)^n ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2018
STATUS
approved