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A354654
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G.f. A(x) satisfies: -4 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).
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9
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5, 124, 9300, 912520, 102616748, 12498655200, 1604505393140, 213790010204692, 29287693334340840, 4099332312599011100, 583685111605968443456, 84277588096627459702860, 12310921909740521584887824, 1816058097888803062860159620, 270156262107594683175523302780
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OFFSET
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0,1
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) -4 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) -4 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) -4 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) -4 = Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n-1)*A(x)^n) * (1 - x^n*A(x)^(n-1)), by the Jacobi Triple Product identity.
a(n) = (-1)^(n+1) * Sum_{k=0..2*n+1} A354649(n,k)*(-4)^k, for n >= 0.
a(n) = Sum_{k=0..2*n+1} A354650(n,k)*4^k, for n >= 0.
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EXAMPLE
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G.f.: A(x) = 5 + 124*x + 9300*x^2 + 912520*x^3 + 102616748*x^4 + 12498655200*x^5 + 1604505393140*x^6 + 213790010204692*x^7 + 29287693334340840*x^8 + ...
such that A = A(x) satisfies:
(1) -4 = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ...
(2) -4 = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
(3) -4 = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ...
(4) -4 = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ...
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PROG
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(PARI) {a(n) = my(A=[5]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(4 + sum(m=0, sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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