OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
EXAMPLE
G.f. A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 106*x^5 + 552*x^6 + 3154*x^7 + 19354*x^8 + 125787*x^9 + 857793*x^10 + 6097668*x^11 + 44974804*x^12 + ...
such that the following series are equal:
B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + x^9*A(x)^45 + ...
B(x) = 1 + x*(1+x*A(x)) + x^2*(1+x*A(x))^4 + x^3*(1+x*A(x))^9 + x^4*(1+x*A(x))^16 + x^5*(1+x*A(x))^25 + x^6*(1+x*A(x))^36 + x^7*(1+x*A(x))^49 + ...
where
B(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 92*x^5 + 426*x^6 + 2140*x^7 + 11517*x^8 + 65804*x^9 + 396499*x^10 + 2506624*x^11 + 16559723*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(m=0, #A, x^m*( Ser(A)^(m*(m+1)/2) - (1+x*Ser(A))^(m^2) ) ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 26 2019
STATUS
approved