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A194042
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G.f. satisfies: A(x) = ( Sum_{n>=0} q^(n*(n+1)/2) )^8 where q=x*A(x).
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3
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1, 8, 92, 1248, 18590, 294032, 4848456, 82433472, 1434755717, 25438412696, 457838000316, 8342826818080, 153615385821902, 2853694780131056, 53419650732453288, 1006670308475537216, 19081647909610147674, 363577167520213046504, 6959630216092660689968
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OFFSET
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0,2
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LINKS
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FORMULA
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Let q = x*A(x), then g.f. A(x) satisfies:
(1) A(x) = ( Sum_{n>=0} (2*n+1) * q^n/(1 - q^(2*n+1)) )^2,
(2) A(x) = Sum_{n>=0} (n+1)^3 * q^n/(1 - q^(2*n+2)),
(3) A(x) = Product_{n>=1} (1 + q^n)^8*(1 - q^(2*n))^8,
(4) A(x) = exp( Sum_{n>=1} 8*(q^n/(1 + q^n))/n ),
(5) A(x/F(x)^8) = F(x)^8 where F(x) = Sum_{n>=0} x^(n*(n+1)/2),
due to q-series identities.
a(n) ~ c * d^n / n^(3/2), where d = 20.757466132085824914671628173626682246438530051407107800521045715415164484946... and c = 1.12762897595401376103508613947431975160374771449653856610031074284717... - Vaclav Kotesovec, Oct 07 2020
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EXAMPLE
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G.f.: A(x) = 1 + 8*x + 92*x^2 + 1248*x^3 + 18590*x^4 + 294032*x^5 +...
where
(0) A(x)^(1/8) = 1 + x*A(x) + x^3*A(x)^3 + x^6*A(x)^6 + x^10*A(x)^10 + x^15*A(x)^15 + x^21*A(x)^21 +...
(1) A(x)^(1/2) = 1/(1-x*A(x)) + 3*x*A(x)/(1-x^3*A(x)^3) + 5*x^2*A(x)^2/(1-x^5*A(x)^5) + 7*x^3*A(x)^3/(1-x^7*A(x)^7) +...
(2) A(x) = 1/(1-x^2*A(x)^2) + 8*x*A(x)/(1-x^4*A(x)^4) + 27*x^2*A(x)^2/(1-x^6*A(x)^6) + 64*x^3*A(x)^3/(1-x^8*A(x)^8) +...
(3) A(x) = (1+x*A(x))^8*(1-x^2*A(x)^2)^8 * (1+x^2*A(x)^2)^8*(1-x^4*A(x)^4)^8 * (1+x^3*A(x)^3)^8*(1-x^6*A(x)^6)^8 * (1+x^4*A(x)^4)^8*(1-x^8*A(x)^8)^8 *...
(4) log(A(x)) = 8*x*A(x)/(1+x*A(x)) + 8*(x^2*A(x)^2/(1+x^2*A(x)^2))/2 + 8*(x^3*A(x)^3/(1+x^3*A(x)^3))/3 + 8*(x^4*A(x)^4/(1+x^4*A(x)^4))/4 +...
Related expansions begin:
_ A(x)^(1/8) = 1 + x + 8*x^2 + 93*x^3 + 1272*x^4 + 19058*x^5 + 302705*x^6 + 5007234*x^7 + 85341048*x^8 +...+ A194043(n)*x^n +...
_ A(x)^(1/2) = 1 + 4*x + 38*x^2 + 472*x^3 + 6685*x^4 + 102340*x^5 + 1649446*x^6 + 27574712*x^7 + 473750970*x^8 +...+ A194044(n)*x^n +...
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MATHEMATICA
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(* Calculation of constant d: *) 1/r /. FindRoot[{2*s^(1/8) == QPochhammer[-1, r*s]*QPochhammer[r^2*s^2, r^2*s^2], r*s*Derivative[0, 1][QPochhammer][-1, r*s] / QPochhammer[-1, r*s] + r^2*s^(15/8) * QPochhammer[-1, r*s] * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] - 2*Log[1 - r^2*s^2]/Log[r^2*s^2] - 2*QPolyGamma[0, 1, r^2*s^2]/Log[r^2*s^2] == 1/8}, {r, 1/20}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 17 2024 *)
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PROG
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(PARI) {a(n)=local(A=1+x, T=sum(m=0, sqrtint(2*n+1), x^(m*(m+1)/2))+x*O(x^n)); A=(serreverse(x/T^8)/x); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, sqrt(2*n+1), (x*A+x*O(x^n))^(m*(m+1)/2))^8); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (2*m+1)*(x*A)^m/(1-(x*A+x*O(x^n))^(2*m+1)))^2); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (m+1)^3*(x*A)^m/(1-(x*A+x*O(x^n))^(2*m+2)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+(x*A)^m)*(1-(x*A)^(2*m)+x*O(x^n)))^8); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, 8*(x*A)^m/(1+(x*A)^m+x*O(x^n))/m))); polcoeff(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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