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A370142
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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).
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5
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1, -3, 9, -27, 78, -219, 591, -1500, 3420, -6153, 3315, 44466, -324276, 1627002, -7069893, 28345875, -107618916, 391749108, -1375531170, 4669215090, -15311251593, 48316101369, -145501913850, 411323278248, -1053727809204, 2226156968586, -2433380638410, -10543933246791
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
Let Q(x) = 1 + 3*Sum_{n>=1} x^(n*(n+1)/2), then
(1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + 2*A(x)).
(2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * 2*A(x)).
(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.
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EXAMPLE
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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 + ...
Let Q(x) = 1 + 3*Sum_{n>=1} x^(n*(n+1)/2)
then A = A(x) satisfies
(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) + ...
also
(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)) + ...
Further, A = A(x) satisfies the continued fraction given by
(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 - ...)))))))).
where
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) + ...
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PROG
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(PARI) {a(n, y=2) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
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]}
for(n=1, 40, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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