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A268301
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G.f. satisfies: -1 = Product_{n>=1} (1-x^n) * (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where g.f. A(x) = Sum_{n>=0} a(n)/2*(x/4)^n.
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4
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1, -7, -70, -795, -13802, -277782, -6093708, -139376659, -3297234754, -79988099074, -1979248977748, -49758116194846, -1267321717299236, -32631825106297228, -848030793254951704, -22214311484843607811, -585938143786366837938, -15548874443787002057610, -414829266882771282611204, -11120089118043870668697578, -299364678394845043715844268, -8090271856987498430846360564
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OFFSET
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0,2
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COMMENTS
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The g.f. utilizes the Jacobi Triple Product: Product_{n>=1} (1-x^n)*(1 - x^n*a)*(1 - x^(n-1)/a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * a^n.
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LINKS
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FORMULA
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Given g.f. A(x) = Sum_{n>=0} a(n)/2 * (x/4)^n, then g.f. also satisfies:
(1) -1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n,
(2) A(x) = Product_{n>=1} (1-x^n) * (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)),
(3) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
(4) x = Sum_{n>=1} A268299(n) * x^n * A(x)^n.
a(n) is odd iff n = 2^k-1 for k>=0 (conjecture).
a(n) ~ -c * d^n / n^(3/2), where d = 29.101591090693617170487962339050658... and c = 0.1385938593465955724446602611055779... . - Vaclav Kotesovec, Mar 02 2016
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EXAMPLE
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G.f.: A(x) = 1/2 - 7/2*x/4 - 70/2*x^2/4^2 - 795/2*x^3/4^3 - 13802/2*x^4/4^4 - 277782/2*x^5/4^5 - 6093708/2*x^6/4^6 - 139376659/2*x^7/4^7 - 3297234754/2*x^8/4^8 - 79988099074/2*x^9/4^9 - 1979248977748/2*x^10/4^10 -...
where g.f. A(x) satisfies the Jacobi Triple Product:
-1 = (1-x)*(1-x*A(x))*(1-1/A(x)) * (1-x^2)*(1-x^2*A(x))*(1-x/A(x)) * (1-x^3)*(1-x^3*A(x))*(1-x^2/A(x)) * (1-x^4)*(1-x^4*A(x))*(1-x^3/A(x)) * (1-x^5)*(1-x^5*A(x))*(1-x^4/A(x)) * (1-x^6)*(1-x^6*A(x))*(1-x^5/A(x)) *...
also
A(x) = (1-x)*(1-x/A(x))*(1-A(x)) * (1-x^2)*(1-x^2/A(x))*(1-x*A(x)) * (1-x^3)*(1-x^3/A(x))*(1-x^2*A(x)) * (1-x^4)*(1-x^4/A(x))*(1-x^3*A(x)) * (1-x^5)*(1-x^5/A(x))*(1-x^4*A(x)) * (1-x^6)*(1-x^6/A(x))*(1-x^5*A(x)) *...
RELATED SERIES.
1/A(x) = 2 + 7*2*x/4 + 119*2*x^2/4^2 + 2118*2*x^3/4^3 + 42523*2*x^4/4^4 + 914922*2*x^5/4^5 + 20745494*2*x^6/4^6 + 487390092*2*x^7/4^7 + 11764545555*2*x^8/4^8 + 289962708802*2*x^9/4^9 +...+ A268300(n)*2*x^n/4^n +...
Series_Reversion( x*A(x) ) = 2*x + 7*x^2 + 84*x^3 + 1240*x^4 + 20942*x^5 + 382344*x^6 + 7354688*x^7 + 146810440*x^8 + 3012778758*x^9 + 63167322872*x^10 +...+ A268299(n)*x^n +..., an integer series.
Let J(x) = Sum_{n>=1} x^(n*(n-1)/2) * (A(x)^(n-1) + 1/A(x)^n),
then J(x) is an integer series:
J(x) = 3 + 8*x + 28*x^2 + 144*x^3 + 736*x^4 + 4024*x^5 + 22912*x^6 + 134784*x^7 + 813476*x^8 + 5010904*x^9 + 31379808*x^10 +..+ A268302(n)*x^n +...
and J(x) = Product_{n>=1} (1-x^n) * (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)).
Conjecture: Product_{n>=1} (1-x^n) * (1 + k*x^n*A(x)) * (1 + k*x^(n-1)/A(x)) yields an integer series for all integer k.
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PROG
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(PARI) {a(n) = my(A=1/2+x, t=floor(sqrt(2*n+1)+1/2)); for(i=0, n, A = (A + sum(m=-t, t, x^(m*(m-1)/2) * (-A)^m +x*O(x^n)) )/2 ); 2*4^n * polcoeff(A, n)}
for(n=0, 30, 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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