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A326564
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O.g.f. A(x) satisfies: 0 = Sum_{n>=1} (b(n) - A(x))^n * (2*x)^n / n, where b(n) = 1 if n is odd or b(n) = 2 if n is even.
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2
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1, 1, -2, 7, -26, 102, -420, 1787, -7794, 34666, -156636, 716982, -3317700, 15494156, -72935624, 345701843, -1648489762, 7902956738, -38067806892, 184152092450, -894259126540, 4357738501844, -21302682030328, 104439435098718, -513390992000340, 2529846489669412, -12494572784556440, 61838364112438732, -306647601790749384, 1523380558254732312, -7580755340625743760, 37783723921640161923
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OFFSET
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0,3
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COMMENTS
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a(n) is odd iff n = 2^k - 1 for k >= 0.
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LINKS
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FORMULA
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O.g.f. A = A(x) satisfies:
(1) 0 = Sum_{n>=1} (3 + (-1)^n - 2*A(x))^n * x^n / n.
(2) 0 = arctanh(2*x - 2*x*A) - log(1 - 4*x^2*(2 - A)^2)/2.
(3) 1 - 4*x^2*(2 - A)^2 = (1 + 2*x - 2*x*A) / (1 - 2*x + 2*x*A).
(4) A(x) = 1 + (A - 2)^2*x + 2*(A - 1)*(A - 2)^2*x^2.
(5) 0 = 2*(A - 1)*(A - 2)^2*x^2 + (A - 2)^2*x - (A - 1).
(6) x = ( sqrt( (A-2)^4 + 8*(A-1)^2*(A-2)^2 ) - (A-2)^2 ) / (4*(A-1)*(A-2)^2).
(7) A(x) = 2 - (1/x) * Series_Reversion( x + x^2/(1 - 2*x^2) ).
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EXAMPLE
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O.g.f.: A(x) = 1 + x - 2*x^2 + 7*x^3 - 26*x^4 + 102*x^5 - 420*x^6 + 1787*x^7 - 7794*x^8 + 34666*x^9 - 156636*x^10 + 716982*x^11 - 3317700*x^12 + 15494156*x^13 - 72935624*x^14 + 345701843*x^15 - 1648489762*x^16 + ...
such that
0 = (1 - A(x))*(2*x) + (2 - A(x))^2*(2*x)^2/2 + (1 - A(x))^3*(2*x)^3/3 + (2 - A(x))^4*(2*x)^4/4 + (1 - A(x))^5*(2*x)^5/5 + (2 - A(x))^6*(2*x)^6/6 + (1 - A(x))^7*(2*x)^7/7 + (2 - A(x))^8*(2*x)^8/8 + (1 - A(x))^9*(2*x)^9/9 + ...
SPECIAL ARGUMENTS.
A( (3 - sqrt(17))/6 ) = 1/2.
A( (15 - sqrt(513))/40 ) = 1/3.
ODD TERMS.
The odd numbers occur at positions 2^n-1 and begin
[1, 1, 7, 1787, 345701843, 37783723921640161923, 1297226675901009799785880946943488094880739, 4359630365907394639251834255689265800511483817161978056491648421720696612963282942355107, ...].
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PROG
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(PARI) /* By definition */
{a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, ( ((m+1)%2) + 1 - Ser(A) )^m * (2*x)^m/m), #A)/2); A[n+1]}
for(n=0, 32, print1(a(n), ", "))
(PARI) /* From: A(x) = 2 - (1/x) * Series_Reversion( x + x^2/(1 - 2*x^2) ) */
{a(n) = my(A = 2 - (1/x)*serreverse(x + x^2/(1 - 2*x^2 +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 32, 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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