OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..250
FORMULA
G.f. A(x) allows the following sums to be equal:
(1) B(x) = Sum_{n>=0} A(x)^(n*(n-1)+1) * x^n.
(2) B(x) = Sum_{n>=0} (A(x)^(n-1) + 1)^n * x^n.
(3) B(x) = Sum_{n>=0} A(x)^(n*(n-1)) * x^n / (1 - x*A(x)^n)^(n+1).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 134*x^5 + 775*x^6 + 4856*x^7 + 32359*x^8 + 226688*x^9 + 1656745*x^10 + 12566075*x^11 + 98550684*x^12 + ...
such that the following sums are equal
B(x) = A(x) + A(x)*x + A(x)^3*x^2 + A(x)^7*x^3 + A(x)^13*x^4 + A(x)^21*x^5 + A(x)^31*x^6 + A(x)^43*x^7 + A(x)^57*x^8 + ... + A(x)^(n*(n-1)+1)*x^n + ...
and
B(x) = 1 + 2*x + (1 + A(x))^2*x^2 + (1 + A(x)^2)^3*x^3 + (1 + A(x)^3)^4*x^4 + (1 + A(x)^4)^5*x^5 + (1 + A(x)^5)^6*x^6 + ... + (1 + A(x)^(n+1))^n*x^n + ...
also
B(x) = 1/(1 - x) + x/(1 - x*A(x))^2 + A(x)^2*x^2/(1 - x*A(x)^2)^3 + A(x)^6*x^3/(1 - x*A(x)^3)^4 + ... + A(x)^(n*(n-1))*x^n/(1 - x*A(x)^n)^(n+1) + ...
where
B(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 49*x^4 + 240*x^5 + 1328*x^6 + 8014*x^7 + 51691*x^8 + 351839*x^9 + 2505762*x^10 + 18563322*x^11 + 142460948*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^(m-1) + 1)^m*x^m - Ser(A)^(m^2-m+1)*x^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 23 2019
STATUS
approved