A196354
E.g.f.: 1 + Sum_{n>=1} 2*cosh(n*x) * x^(n^2).
Original entry on oeis.org
1, 2, 0, 6, 48, 10, 2880, 14, 53760, 725778, 645120, 359251222, 6082560, 42032390426, 49201152, 2648040595230, 41845937602560, 115757203161634, 102437981698129920, 3958896348126758, 51901909523009372160, 113368395423628842, 12788630502806158049280
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 0*x^2/2! + 6*x^3/3! + 48*x^4/4! + 10*x^5/5! + 2880*x^6/6! + 14*x^7/7! + 53760*x^8/8! +...
The e.g.f. A(x) may be expressed by the series:
A(x) = 1 + 2*cosh(x)*x + 2*cosh(2*x)*x^4 + 2*cosh(3*x)*x^9 + 2*cosh(4*x)*x^16 + 2*cosh(5*x)*x^25 +...
and by Jacobi's triple product:
A(x) = (1-x^2)*(1+x*exp(x))*(1+x/exp(x)) * (1-x^4)*(1+x^3*exp(x))*(1+x^3/exp(x)) * (1-x^6)*(1+x^5*exp(x))*(1+x^5/exp(x)) * (1-x^8)*(1+x^7*exp(x))*(1+x^7/exp(x)) *...
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a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (Series[ EllipticTheta[ 3, x I/2, q], {q, 0, n}] // Normal // TrigToExp) /. {x -> q}, {q, 0, n}]] (* Michael Somos, Nov 18 2011 *)
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{a(n)=local(A=1+x); A=1+sum(m=1,sqrtint(n+1), 2*cosh(m*x+x*O(x^n))*x^(m^2)); n!*polcoeff(A, n)}
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/* By Jacobi's Triple Product Identity: */
{a(n)=local(A=1+x); A=prod(m=1, n\2+1, (1-x^(2*m))*(1+exp(x+x*O(x^n))*x^(2*m-1))*(1+exp(-x+x*O(x^n))*x^(2*m-1)+x*O(x^n))); n!*polcoeff(A, n)}
A200222
G.f. satisfies: A(x) = 1 + Sum_{n>=1} (x*A(x))^(n^2) * (A(x)^n + 1/A(x)^n).
Original entry on oeis.org
1, 2, 4, 12, 42, 164, 688, 3024, 13680, 63110, 295520, 1401012, 6713280, 32470468, 158343504, 777725264, 3843992546, 19104857608, 95419519076, 478668009828, 2410698765472, 12184259877320, 61782045169312, 314202878599696, 1602270787137472, 8191160756085318, 41971595130249968, 215522156779513584
Offset: 0
G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 42*x^4 + 164*x^5 + 688*x^6 +...
The g.f. A = A(x) may be expressed by the series:
A(x) = 1 + x*A*(A + 1/A) + x^4*A^4*(A^2 + 1/A^2) + x^9*A^9*(A^3 + 1/A^3) + x^16*A^16*(A^4 + 1/A^4) + x^25*A^25*(A^5 + 1/A^5) +...
and by the Jacobi triple product:
A(x) = (1+x)*(1+x*A^2)*(1-x^2*A^2) * (1+x^3*A^2)*(1+x^3*A^4)*(1-x^4*A^4) * (1+x^5*A^4)*(1+x^5*A^6)*(1-x^6*A^6) * (1+x^7*A^6)*(1+x^7*A^8)*(1-x^8*A^8) *...
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(* Calculation of constant d: *) 1/r /. FindRoot[{s == r^2*s^2 * QPochhammer[-1/r, r^2*s^2] * QPochhammer[-1/(r*s^2), r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]/((1 + r)*(1 + r*s^2)), 1 - r*s^2 - 2*(1 + r*s^2) * QPolyGamma[0, 1, r^2*s^2] / Log[r^2*s^2] + 2*(1 + r*s^2) * QPolyGamma[0, Log[-1/(r*s^2)] / Log[r^2*s^2], r^2*s^2] / Log[r^2*s^2] + 2*r^2*s^2*((1 + r*s^2)*(Derivative[0, 1][QPochhammer][-1/r, r^2*s^2] / QPochhammer[-1/r, r^2*s^2] + Derivative[0, 1][QPochhammer][-1/(r*s^2), r^2*s^2] / QPochhammer[-1/(r*s^2), r^2*s^2]) + r^2*s * QPochhammer[-1/r, r^2*s^2] * QPochhammer[-1/(r*s^2), r^2*s^2] * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] / (1 + r)) == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)
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{a(n)=local(A=1+x); for(i=1,n,A=1+sum(m=1, sqrtint(n+1), (A^m +1/(A+x*O(x^n))^m)*(x*A)^(m^2))); polcoeff(A, n)}
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/* By Jacobi's Triple Product Identity: */
{a(n)=local(A=1+x); for(i=1,n,A=prod(m=1, n\2+1, (1+x^(2*m-1)*A^(2*m-2)+x*O(x^n))*(1+x^(2*m-1)*A^(2*m))*(1-x^(2*m)*A^(2*m)) )); polcoeff(A, n)}
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