OFFSET
0,2
FORMULA
Given g.f. A(x), the following sums are all equal:
(1) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(n+4)),
(2) B(x) = Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^(5*n+3)),
(3) B(x) = Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^(5*n+2)),
(4) B(x) = Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^(n+1)),
(5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+5*n) * (1 - x^2*A(x)^(2*n+5)) / ((1 - x*A(x)^(n+1))*(1 - x*A(x)^(n+4))),
(6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(5*n^2+5*n) * (1 - x^2*A(x)^(10*n+5)) / ((1 - x*A(x)^(5*n+2))*(1 - x*A(x)^(5*n+3)));
see the example section for the value of B(x).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 24*x^2 + 460*x^3 + 10480*x^4 + 262264*x^5 + 6968920*x^6 + 193069480*x^7 + 5515713624*x^8 + 161332352440*x^9 + 4807707617064*x^10 + ...
such that
B(x) = 1/(1-x*A(x)^4) + x*A(x)/(1 - x*A(x)^5) + x^2*A(x)^2/(1 - x*A(x)^6) + x^3*A(x)^3/(1 - x*A(x)^7) + x^4*A(x)^4/(1 - x*A(x)^8) + ...
and
B(x) = 1/(1-x*A(x)^3) + x*A(x)^2/(1 - x*A(x)^8) + x^2*A(x)^4/(1 - x*A(x)^13) + x^3*A(x)^6/(1 - x*A(x)^18) + x^4*A(x)^8/(1 - x*A(x)^23) + ...
also
B(x) = 1/(1-x*A(x)^2) + x*A(x)^3/(1 - x*A(x)^7) + x^2*A(x)^6/(1 - x*A(x)^12) + x^3*A(x)^9/(1 - x*A(x)^17) + x^4*A(x)^12/(1 - x*A(x)^22) + ...
further
B(x) = 1/(1-x*A(x)) + x*A(x)^4/(1 - x*A(x)^2) + x^2*A(x)^8/(1 - x*A(x)^3) + x^3*A(x)^12/(1 - x*A(x)^4) + x^4*A(x)^16/(1 - x*A(x)^5) + ...
where
B(x) = 1 + 2*x + 13*x^2 + 180*x^3 + 3541*x^4 + 81806*x^5 + 2066157*x^6 + 55253320*x^7 + 1537808581*x^8 + 44084046546*x^9 + 1292851214781*x^10 + ...
PROG
(PARI) {a(n) = my(A=[1, 2]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
H[#A-1] = -polcoeff( sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(m+4)) ) - sum(m=0, #A, x^m*Ser(A)^(2*m)/(1 - x*Ser(A)^(5*m+3)) ), #A)/4; A=H); A[n+1] }
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 10 2021
STATUS
approved