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%I #12 Feb 15 2024 08:20:24
%S 1,-3,9,-27,78,-219,591,-1500,3420,-6153,3315,44466,-324276,1627002,
%T -7069893,28345875,-107618916,391749108,-1375531170,4669215090,
%U -15311251593,48316101369,-145501913850,411323278248,-1053727809204,2226156968586,-2433380638410,-10543933246791
%N Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + 2*A(x)) = 1 + 3*Sum_{n>=1} x^(n*(n+1)/2).
%H Paul D. Hanna, <a href="/A370142/b370142.txt">Table of n, a(n) for n = 1..210</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F Let Q(x) = 1 + 3*Sum_{n>=1} x^(n*(n+1)/2), then
%F (1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + 2*A(x)).
%F (2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * 2*A(x)).
%F (3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + 2*A(x))/(1 + x^n + 2*A(x) - F(n+1)), a continued fraction.
%e G.f.: A(x) = x - 3*x^2 + 9*x^3 - 27*x^4 + 78*x^5 - 219*x^6 + 591*x^7 - 1500*x^8 + 3420*x^9 - 6153*x^10 + 3315*x^11 + 44466*x^12 + ...
%e Let Q(x) = 1 + 3*Sum_{n>=1} x^(n*(n+1)/2)
%e then A = A(x) satisfies
%e (1) Q(x) = 1 + (x + 2*A) + (x + 2*A)*(x^2 + 2*A) + (x + 2*A)*(x^2 + 2*A)*(x^3 + 2*A) + (x + 2*A)*(x^2 + 2*A)*(x^3 + 2*A)*(x^4 + 2*A) + (x + 2*A)*(x^2 + 2*A)*(x^3 + 2*A)*(x^4 + 2*A)*(x^5 + 2*A) + ...
%e also
%e (2) Q(x) = 1/(1 - 2*A) + x/((1 - 2*A)*(1 - x*2*A)) + x^3/((1 - 2*A)*(1 - x*2*A)*(1 - x^2*2*A)) + x^6/((1 - 2*A)*(1 - x*2*A)*(1 - x^2*2*A)*(1 - x^3*2*A)) + x^10/((1 - 2*A)*(1 - x*2*A)*(1 - x^2*2*A)*(1 - x^3*2*A)*(1 - x^4*2*A)) + ...
%e Further, A = A(x) satisfies the continued fraction given by
%e (3) Q(x) = 1/(1 - (x + 2*A)/(1 + x + 2*A - (x^2 + 2*A)/(1 + x^2 + 2*A - (x^3 + 2*A)/(1 + x^3 + 2*A - (x^4 + 2*A)/(1 + x^4 + 2*A - (x^5 + 2*A)/(1 + x^5 + 2*A - (x^6 + 2*A)/(1 + x^6 + 2*A - (x^7 + 2*A)/(1 - ...)))))))).
%e where
%e Q(x) = 1 + 3*x + 3*x^3 + 3*x^6 + 3*x^10 + 3*x^15 + 3*x^21 + ... + 3*x^(n*(n+1)/2) + ...
%o (PARI) {a(n,y=2) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A[#A] = polcoeff( (sum(m=1,#A, prod(k=1,m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1,sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1) ); H=A; A[n+1]}
%o for(n=1,40, print1(a(n),", "))
%Y Cf. A370140, A370141, A370143, A370144.
%K sign
%O 1,2
%A _Paul D. Hanna_, Feb 14 2024
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