OFFSET
0,5
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 16*x^5 + 75*x^6 + 332*x^7 + 1509*x^8 + 7134*x^9 + 34733*x^10 + 173318*x^11 + 885600*x^12 + ...
where the following sums are equal:
B(x) = 1 + A(x)*x + A(x)^3*x^2 + A(x)^6*x^3 + A(x)^10*x^4 + A(x)^15*x^5 + A(x)^21*x^6 + A(x)^28*x^7 + ... + A(x)^(n*(n+1)/2)*x^n + ...
also
B(x) = A(x) + A(x)^4*x^2 + A(x)^9*x^4 + A(x)^16*x^6 + A(x)^25*x^8 + A(x)^36*x^10 + A(x)^49*x^12 + ... + A(x)^((n+1)^2) * x^(2*n) + ...
where
B(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 45*x^5 + 164*x^6 + 653*x^7 + 2767*x^8 + 12313*x^9 + 57053*x^10 + 273493*x^11 + 1350298*x^12 + 6846378*x^13 + 35577209*x^14 + 189216369*x^15 + 1029024489*x^16 + 5719246977*x^17 + 32477483024*x^18 + 188419488119*x^19 + 1116859062391*x^20 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A, x^n * Ser(A)^(n*(n+1)/2) ) - sum(n=0, #A, x^(2*n) * Ser(A)^((n+1)^2) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 03 2020
STATUS
approved