M. Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

OEIS Server: https://oeis.org/edit/global/2897

M. Somos, <a href="http://somos.crg4.com="/A010815/multiqa010815.htmltxt">Introduction to Ramanujan theta functions</a>

OEIS Server: https://oeis.org/edit/global/2832

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G. C. Greubel, <a href="/A129451/b129451.txt">Table of n, a(n) for n = 0..10000</a>

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Expansion of psif(-q)* phi(qx, -x^3)/ chi(q) f(-x, x^2) in powers of qx where phi(), psi(), chi() aref(, ) is Ramanujan's general theta functionsfunction.

1, -2, 2, -2, 1, -2, 2, -2, 3, 0, 2, -2, 2, -2, 0, -4, 2, -2, 2, 0, 1, -2, 4, -2, 0, -2, 2, -2, 3, -2, 2, 0, 2, -2, 0, -2, 4, -2, 2, 0, 2, -4, 0, -4, 0, -2, 2, -2, 1, 0, 4, -2, 2, 0, 2, -2, 2, -4, 2, 0, 3, -2, 2, -2, 0, 0, 2, -4, 2, 0, 2, -4, 2, -2, 0, 0, 2, -2, 4, -2, 4, -2, 0, -2, 0, -4, 0, -2, 1, 0, 2, -2, 4, -4, 0, -2, 2, 0, 4, 0, 2, -2, 2, -2, 1

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)

M. Somos, <a href="http://cissomos.csuohiocrg4.edu/~somoscom/multiq.pdfhtml">Introduction to Ramanujan theta functions</a>

Expansion of q^(-1/6)* eta(q)^2* eta(q^4)^2* etapsi(-x) * phi(qx^6)^5/ (eta3) / chi(qx) = f(-x^2)^3* eta * phi(qx^3)^2* eta) / f(q^12x)^2) in powers of x where phi(), psi(), chi() are Ramanujan theta qfunctions.

Expansion of q^(-1/6) * eta(q)^2 * eta(q^4)^2 * eta(q^6)^5 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^2) in powers of q.

a(n)= ) = b(6n+6*n + 1) where b(n) () is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 5, 11 (mod 12), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (-1)^e* (e+1) if p == 7 (mod 12).

a(n) = A129449(3*n).

qG.f. = 1 - 2*q^7x + 2*qx^132 - 2*qx^193 + qx^254 - 2*qx^315 + 2*qx^376 - 2*qx^437 + 3*x^8 + 2*qx^49 +...10 + ...

G.f. = q - 2*q^7 + 2*q^13 - 2*q^19 + q^25 - 2*q^31 + 2*q^37 - 2*q^43 + 3*q^49 + ...

a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 1}, DivisorSum[ m, KroneckerSymbol[ -4, #] KroneckerSymbol[ 12, m/#] &]]]; (* Michael Somos, Nov 11 2015 *)

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] QPochhammer[ x^2]^3 / QPochhammer[ -x]^2, {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-1/8) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 3, 0, x^3] QPochhammer[ x, -x], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

(PARI) {a(n)= ) = if(( n<0, 0, n= = 6*n+ + 1; sumdiv(n, , d, kronecker(-( -4, , d)*) * kronecker(( 12, , n/d)))})))};

(PARI) {a(n)= local) = my(A); if(( n<0, 0, A= = x* * O(x^n); polcoeff( eta(x+ + A)^2* * eta(x^4+ + A)^2* * eta(x^6+ + A)^5/ ( / (eta(x^2+ + A)^3* * eta(x^3+ + A)^2* * eta(x^12+ + A)^2), n))}))};

Cf. A129449(3n)= a(n).

Cf. A129449.

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Michael Somos: Added more info. Light and space edits. Better name. Cut sequence terms to 260 chars max. Revised Ramanujan theta comment. Updated URL.

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

OEIS Server: https://oeis.org/edit/global/2357

_Michael Somos, _, Apr 16 2007

OEIS Server: https://oeis.org/edit/global/2178

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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