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A370143
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Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + 3*A(x)) = 1 + 4*Sum_{n>=1} x^(n*(n+1)/2).
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5
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1, -4, 16, -64, 252, -980, 3752, -14076, 51384, -180488, 597812, -1788936, 4284828, -3665376, -47694524, 475585724, -3186717720, 18465627936, -98993741736, 504886869432, -2484393916472, 11887606679816, -55566604265244, 254411449305096, -1142492183274444, 5033985939170544
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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 + 4*Sum_{n>=1} x^(n*(n+1)/2), then
(1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + 3*A(x)).
(2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * 3*A(x)).
(3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + 3*A(x))/(1 + x^n + 3*A(x) - F(n+1)), a continued fraction.
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EXAMPLE
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G.f.: A(x) = x - 4*x^2 + 16*x^3 - 64*x^4 + 252*x^5 - 980*x^6 + 3752*x^7 - 14076*x^8 + 51384*x^9 - 180488*x^10 + 597812*x^11 - 1788936*x^12 + ...
Let Q(x) = 1 + 4*Sum_{n>=1} x^(n*(n+1)/2)
then A = A(x) satisfies
(1) Q(x) = 1 + (x + 3*A) + (x + 3*A)*(x^2 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A)*(x^4 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A)*(x^4 + 3*A)*(x^5 + 3*A) + ...
also
(2) Q(x) = 1/(1 - 3*A) + x/((1 - 3*A)*(1 - x*3*A)) + x^3/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)) + x^6/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)*(1 - x^3*3*A)) + x^10/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)*(1 - x^3*3*A)*(1 - x^4*3*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
(3) Q(x) = 1/(1 - (x + 3*A)/(1 + x + 3*A - (x^2 + 3*A)/(1 + x^2 + 3*A - (x^3 + 3*A)/(1 + x^3 + 3*A - (x^4 + 3*A)/(1 + x^4 + 3*A - (x^5 + 3*A)/(1 + x^5 + 3*A - (x^6 + 3*A)/(1 + x^6 + 3*A - (x^7 + 3*A)/(1 - ...)))))))).
where
Q(x) = 1 + 4*x + 4*x^3 + 4*x^6 + 4*x^10 + 4*x^15 + 4*x^21 + ... + 4*x^(n*(n+1)/2) + ...
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PROG
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(PARI) {a(n, y=3) = 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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