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Paul D. Hanna, <a href="/A366231/b366231_1.txt">Table of n, a(n) for n = 0..270</a>
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1 = A(x) /(1+x*A(x)^2) - x^2*A(x)^3/((1+x*A(x)^2)*(1+x^3*A(x)^2) ) + x^4*A(x)^5/((1+x*A(x)^2)*(1+x^3*A(x)^2)*(1+x^5*A(x)^2
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Paul D. Hanna, <a href="/A366231/b366231_1.txt">Table of n, a(n) for n = 0..270</a>
Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} (-1)^n * x^n * A(x)^(2*n+1) * Product_{k=1..n} (1 + x^k).
1 = A(x) - x*A(x)^23*(1+x) + x^2*A(x)^35*(1+x)*(1+x^2) - x^3*A(x)^47*(1+x)*(1+x^2)*(1+x^3) + x^4*A(x)^59*(1+x)*(1+x^2)*(1+x^3)*(1+x^4) - x^5*A(x)^611*(1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5) +- ...
allocated for Paul D. Hanna
Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} (-1)^n * x^n * A(x)^(n+1) * Product_{k=1..n} (1 + x^k).
1, 1, 3, 10, 37, 146, 604, 2582, 11319, 50607, 229875, 1057856, 4921427, 23108430, 109370632, 521229470, 2499113258, 12046661239, 58346721541, 283805084926, 1385781218558, 6790201444942, 33377058382130, 164540328122236, 813301767625587, 4029903322301757, 20013362007322192
0,3
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n>=0} (-1)^n * x^n * A(x)^(2*n+1) * Product_{k=1..n} (1 + x^k).
(2) 1 = Sum_{n>=0} (-1)^n * x^(2*n) * A(x)^(2*n+1) / Product_{k=1..n+1} (1 + x^(2*k-1)*A(x)^2).
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 146*x^5 + 604*x^6 + 2582*x^7 + 11319*x^8 + 50607*x^9 + 229875*x^10 + 1057856*x^11 + 4921427*x^12 + ...
where
1 = A(x) - x*A(x)^2*(1+x) + x^2*A(x)^3*(1+x)*(1+x^2) - x^3*A(x)^4*(1+x)*(1+x^2)*(1+x^3) + x^4*A(x)^5*(1+x)*(1+x^2)*(1+x^3)*(1+x^4) - x^5*A(x)^6*(1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5) +- ...
also, by a q-series identity, we have
1 = A(x) - x^2*A(x)^3/(1+x*A(x)^2) + x^4*A(x)^5/((1+x*A(x)^2)*(1+x^3*A(x)^2)) - x^6*A(x)^7/((1+x*A(x)^2)*(1+x^3*A(x)^2)*(1+x^5*A(x)^2)) + x^8*A(x)^9/((1+x*A(x)^2)*(1+x^3*A(x)^2)*(1+x^5*A(x)^2)*(1+x^7*A(x)^2)) -+ ...
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(m=0, #A, (-1)^m * x^m * Ser(A)^(2*m+1) * prod(k=1, m, 1 + x^k) ), #A-1) ); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
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Paul D. Hanna, Nov 26 2023
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editing
allocated for Paul D. Hanna
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