A001934 -id:A001934 - cp's OEIS frontend

A001939 Expansion of (psi(-x) / phi(-x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 5, 20, 65, 185, 481, 1165, 2665, 5820, 12220, 24802, 48880, 93865, 176125, 323685, 583798, 1035060, 1806600, 3108085, 5276305, 8846884, 14663645, 24044285, 39029560, 62755345, 100004806, 158022900, 247710570, 385366265, 595212280, 913040649, 1391449780

Offset: 0

N. J. A. Sloane, Simon Plouffe

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

			1 + 5*x + 20*x^2 + 65*x^3 + 185*x^4 + 481*x^5 + 1165*x^6 + 2665*x^7 + ...
q^5 + 5*q^13 + 20*q^21 + 65*q^29 + 185*q^37 + 481*q^45 + 1165*q^53 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000122, A000700, A010054, A121373, A195861.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8))^5, {q, 0, n}] (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (Product[1 - x^k, {k, 4, n, 4}] / Product[1 - x^k, {k, n}])^5, {x, 0, n}] (* Michael Somos, Sep 24 2011 *)
nn = 4*20; b = Flatten[Table[{5, 5, 5, 0}, {nn/4}]]; CoefficientList[x*Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)
QP = QPochhammer; s = (QP[q^4]/QP[q])^5 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^5, n))} /* Michael Somos, Sep 24 2011 */

Expansion of q^(-5/8) * (eta(q^4) / eta(q))^5 in powers of q. - Michael Somos, Sep 24 2011
Euler transform of period 4 sequence [ 5, 5, 5, 0, ...]. - Michael Somos, Sep 24 2011
G.f.: (Product_{k>0} (1 - x^(4*k)) / (1 - x^k))^5. - Michael Somos, Sep 24 2011
a(n) = (-1)^n * A195861(n). - Michael Somos, Sep 24 2011
a(n) ~ 5^(1/4) * exp(sqrt(5*n/2)*Pi) / (64 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 27 2015

A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k).

Original entry on oeis.org

1, 6, 36, 200, 1038, 5160, 24776, 115632, 527172, 2355998, 10349448, 44783064, 191211512, 806737800, 3367294320, 13918479872, 57020736942, 231697484304, 934399998412, 3742041461976, 14888854356840, 58881590423856, 231542984619720, 905666813058384

Offset: 0

Vaclav Kotesovec, Aug 23 2015

nonn

Convolution of A144067 and A256142.

In general, for m > 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m^k), then a(n) ~ m^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(m^(2*j)-1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO]
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A156616, A261519, A260916, A001934, A015128.

nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(3^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 3^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(3^(2*j)-1)) = 0.0887630729103166089354170592729856346...

A001940 Absolute value of coefficients of an elliptic function.

Original entry on oeis.org

1, 6, 27, 98, 309, 882, 2330, 5784, 13644, 30826, 67107, 141444, 289746, 578646, 1129527, 2159774, 4052721, 7474806, 13569463, 24274716, 42838245, 74644794, 128533884, 218881098, 368859591, 615513678, 1017596115, 1667593666, 2710062756, 4369417452

Offset: 0

N. J. A. Sloane, Simon Plouffe

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]

Cf. A001935, A001936, A001937, A001939, A001941, A092877, A093160.

nn = 4*10; b = Flatten[Table[{6, 6, 6, 0}, {nn/4}]]; CoefficientList[x*Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)
nmax = 40; CoefficientList[Series[Product[((1 - x^(4*k)) / (1 - x^k))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2017 *)

G.f.: Product ( 1 - x^k )^(-c(k)), c(k) = 6, 6, 6, 0, 6, 6, 6, 0, ....
a(n) ~ 3^(1/4) * exp(sqrt(3*n)*Pi) / (128*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^6. - Ilya Gutkovskiy, Dec 04 2017

Extended and corrected by Simon Plouffe

A001941 Absolute values of coefficients of an elliptic function.

Original entry on oeis.org

1, 7, 35, 140, 483, 1498, 4277, 11425, 28889, 69734, 161735, 362271, 786877, 1662927, 3428770, 6913760, 13660346, 26492361, 50504755, 94766875, 175221109, 319564227, 575387295, 1023624280, 1800577849, 3133695747, 5399228149, 9214458260, 15584195428

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]

Cf. A001935, A001936, A001937, A001939, A001940, A092877, A093160.

nn = 4*10; b = Flatten[Table[{7, 7, 7, 0}, {nn/4}]]; CoefficientList[x*Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)
nmax = 40; CoefficientList[Series[Product[((1 - x^(4*k)) / (1 - x^k))^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2017 *)

G.f.: Product ( 1 - x^k )^-c(k), c(k) = 7, 7, 7, 0, 7, 7, 7, 0, ....
a(n) ~ 7^(1/4) * exp(sqrt(7*n/2)*Pi) / (256*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^7. - Ilya Gutkovskiy, Dec 04 2017

A004404 Expansion of 1 / (Sum_{n=-oo..oo} x^(n^2))^3.

Original entry on oeis.org

1, -6, 24, -80, 234, -624, 1552, -3648, 8184, -17654, 36816, -74544, 147056, -283440, 535008, -990912, 1803882, -3232224, 5707624, -9943536, 17106960, -29088352, 48922320, -81438528, 134261584, -219336630, 355242288

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..4000 from Robert Israel)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.

Cf. A001934, A004405-A004425, A015128.

S:= series(1/JacobiTheta3(0,x)^3,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 29 2015

nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

a(n) ~ (-1)^n * 3*exp(Pi*sqrt(3*n)) / (64*n^(3/2)) * (1 - sqrt(3)/(Pi*sqrt(n))). - Vaclav Kotesovec, Aug 18 2015, extended Jan 16 2017

A187053 Expansion of (psi(x^2) / psi(x))^3 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 9, -22, 48, -99, 194, -363, 657, -1155, 1977, -3312, 5443, -8787, 13968, -21894, 33873, -51795, 78345, -117312, 174033, -255945, 373353, -540486, 776848, -1109040, 1573209, -2218198, 3109713, -4335840, 6014123, -8300811, 11402928

Offset: 0

Michael Somos, Mar 06 2011

sign

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

			G.f. = 1 - 3*x + 9*x^2 - 22*x^3 + 48*x^4 - 99*x^5 + 194*x^6 - 363*x^7 + ...
G.f. = q^3 - 3*q^11 + 9*q^19 - 22*q^27 + 48*q^35 - 99*q^43 + 194*q^51 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001937, A029840, A083365.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^3, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

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

Expansion of q^(-3/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^3 in powers of q.
Euler transform of period 4 sequence [-3, 6, -3, 0, ...].
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^3.
Convolution inverse of A029840. Convolution cube of A083365. a(n) = (-1)^n * A001937(n).
a(n) ~ (-1)^n * 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

A195861 Expansion of (psi(x) / phi(x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -5, 20, -65, 185, -481, 1165, -2665, 5820, -12220, 24802, -48880, 93865, -176125, 323685, -583798, 1035060, -1806600, 3108085, -5276305, 8846884, -14663645, 24044285, -39029560, 62755345, -100004806, 158022900, -247710570, 385366265

Offset: 0

Michael Somos, Sep 24 2011

sign

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

			G.f. = 1 - 5*x + 20*x^2 - 65*x^3 + 185*x^4 - 481*x^5 + 1165*x^6 - 2665*x^7 + ...
G.f. = q^5 - 5*q^13 + 20*q^21 - 65*q^29 + 185*q^37 - 481*q^45 + 1165*q^53 - 2665*q^61 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001939, A029842.

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), this sequence (b=5).

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16)^(5/8), {q, 0, n + 5/8}]];
a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^5, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2, x^2] / QPochhammer[ -x, x^2])^5, {x, 0, n}];

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

Expansion of q^(-5/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^5 in powers of q.
Euler transform of period 4 sequence [-5, 10, -5, 0, ...].
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^5.
a(n) = (-1)^n * A001939(n). Convolution inverse of A029842.
a(n) ~ (-1)^n * 5^(1/4) * exp(sqrt(5*n/2)*Pi) / (64 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 27 2015

A002318 Expansion of (1/theta_4(q)^2 -1)/4 in powers of q.

Original entry on oeis.org

1, 3, 8, 19, 42, 88, 176, 339, 633, 1150, 2040, 3544, 6042, 10128, 16720, 27219, 43746, 69483, 109160, 169758, 261504, 399272, 604560, 908248, 1354427, 2005710, 2950544, 4313232, 6267642, 9055856, 13013440, 18603603, 26463168, 37464230

Offset: 1

N. J. A. Sloane

nonn

			q + 3*q^2 + 8*q^3 + 19*q^4 + 42*q^5 + 88*q^6 + 176*q^7 + 339*q^8 + 633*q^9 + ...

J. W. L. Glaisher, "On the Coefficients in the q-series for pi/2K and 2G/pi", Quart J. Pure and Applied Math., 21 (1885), 60-76.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001934.

seq(coeff(convert(series(mul(( 1 - x^k )^(-(2+(k mod 2)*2)),k=1..100),x,100),polynom),x,i)/4,i=1..50); (Pab Ter)

Rest[CoefficientList[ Series[(1/EllipticTheta[4, 0, q]^2 - 1)/4, {q, 0, 34}], q]] (* Jean-François Alcover, Jul 18 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Integrate[ (EllipticK[m] - EllipticE[m]) / (8 Sqrt[1 - m] (Pi/2) q), q], {q, 0, n}]] (* Michael Somos, Jan 24 2012 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A)^4 - 1, n) / 4)} /* Michael Somos, Feb 09 2006 */

Expansion of (eta(q^2)^2 / eta(q)^4 - 1) / 4 in powers of q.

a(n) = A001934(n) / 4.

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 18 2005

A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 8, 0, 1, 8, 24, 32, 14, 0, 1, 10, 40, 80, 76, 24, 0, 1, 12, 60, 160, 234, 168, 40, 0, 1, 14, 84, 280, 552, 624, 352, 64, 0, 1, 16, 112, 448, 1110, 1712, 1552, 704, 100, 0, 1, 18, 144, 672, 2004, 3912, 4896, 3648, 1356, 154, 0, 1, 20, 180, 960, 3346, 7896, 12600, 13120, 8184, 2532, 232, 0

Offset: 0

Ilya Gutkovskiy, Jun 10 2017

			Square array begins:
1,   1,    1,    1,     1,     1,  ...
0,   2,    4,    6,     8,    10,  ...
0,   4,   12,   24,    40,    60,  ...
0,   8,   32,   80,   160,   280,  ...
0,  14,   76,  234,   552,  1110,  ...
0,  24,  168,  624,  1712,  3913,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
Rows n=0-2 give: A000012, A005843, A046092.
Main diagonal gives A270919.
Antidiagonal sums give A299108.
Cf. A122141, A286815.

# JacobiTheta4 is defined in A002448.
A288515Column(k, len) = JacobiTheta4(len, -k)
for k in 0:8 A288515Column(k, 8) |> println end # Peter Luschny, Mar 12 2018

Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A001934.

A319552 Expansion of 1/theta_4(q)^3 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, 6, 24, 80, 234, 624, 1552, 3648, 8184, 17654, 36816, 74544, 147056, 283440, 535008, 990912, 1803882, 3232224, 5707624, 9943536, 17106960, 29088352, 48922320, 81438528, 134261584, 219336630, 355242288, 570675904, 909674688, 1439394192, 2261635168, 3529838208

Offset: 0

Seiichi Manyama, Sep 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), this sequence (b=3), A284286 (b=4), A319553 (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004404, A213384.

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^3))

Convolution inverse of A213384.
a(n) = (-1)^n * A004404(n).
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.
G.f.: Product_{k>=1} ((1 - x^(2k))/(1 - x^k)^2)^3.

cp's OEIS Frontend

A001939 Expansion of (psi(-x) / phi(-x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.

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A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k).

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A001940 Absolute value of coefficients of an elliptic function.

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A001941 Absolute values of coefficients of an elliptic function.

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A004404 Expansion of 1 / (Sum_{n=-oo..oo} x^(n^2))^3.

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A187053 Expansion of (psi(x^2) / psi(x))^3 in powers of x where psi() is a Ramanujan theta function.

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A195861 Expansion of (psi(x) / phi(x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.

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