A085140 -id:A085140 - cp's OEIS frontend

A053254 Coefficients of the '3rd-order' mock theta function nu(q).

1, -1, 2, -2, 2, -3, 4, -4, 5, -6, 6, -8, 10, -10, 12, -14, 15, -18, 20, -22, 26, -29, 32, -36, 40, -44, 50, -56, 60, -68, 76, -82, 92, -101, 110, -122, 134, -146, 160, -176, 191, -210, 230, -248, 272, -296, 320, -350, 380, -410, 446, -484, 522, -566, 612, -660, 715, -772, 830, -896, 966, -1038

Offset: 0

Dean Hickerson, Dec 19 1999

In Watson 1936 the function is denoted by upsilon(q). - Michael Somos, Jul 25 2015

			G.f. = 1 - x + 2*x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 4*x^7 + 5*x^8 + ...

George E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, (Example 6, p. 29).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053253, A053255.

Cf. A058140, A067357, A376628.

Series[Sum[q^(n(n+1))/Product[1+q^(2k+1), {k, 0, n}], {n, 0, 9}], {q, 0, 100}]

/* Continued Fraction Expansion: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 + x^(n-k+1)*(1 - x^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 09 2013

G.f.: nu(q) = Sum_{n >= 0} q^(n*(n+1))/((1+q)*(1+q^3)*...*(1+q^(2*n+1)))
(-1)^n*a(n) = number of partitions of n in which even parts are distinct and if k occurs then so does every positive even number less than k.
G.f.: 1/(1 + x*(1-x)/(1 + x^2*(1-x^2)/(1 + x^3*(1-x^3)/(1 + x^4*(1-x^4)/(1 + x^5*(1-x^5)/(1 + ...)))))), a continued fraction. - Paul D. Hanna, Jul 09 2013
a(2*n) = A085140(n). a(2*n + 1) = - A053253(n). - Michael Somos, Jul 25 2015
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jun 15 2019
From Peter Bala, Jan 03 2025: (Start)
a(n) = (-1)^n * A067357(n).
nu(-q) = Sum_{n >= 0} q^n * (1 + q)*(1 + q^3)*...*(1 + q^(2*n-1)) (Andrews, p. 29: in  Example 6 take x = q and y = -q).
Conjecture: a(n) = (-1)^n * (A376628(n) + A376628(n+1)), or equivalently, (1 + q * nu(-q))/(1 + q) = Sum_{n >= 0} q^(n*(n+1))/((1 - q)*(1 - q^3)*...*(1 - q^(2*n-1))), the g.f. of A376628. (End)

A132970 Expansion of phi(-x) * chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 2, -1, 5, -5, 3, -5, 6, -10, 10, -8, 13, -15, 15, -16, 23, -27, 25, -30, 35, -40, 42, -45, 55, -66, 68, -70, 86, -95, 100, -110, 125, -141, 150, -161, 185, -207, 215, -235, 266, -293, 310, -335, 375, -410, 438, -470, 521, -575, 610, -653, 725, -785, 835, -900, 983, -1070, 1140, -1220, 1331

Offset: 0

Michael Somos, Sep 04 2007

sign

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

			G.f. = 1 - 3*x + 2*x^2 - x^3 + 5*x^4 - 5*x^5 + 3*x^6 - 5*x^7 + 6*x^8 + ...
G.f. = 1/q - 3*q^23 + 2*q^47 - q^71 + 5*q^95 - 5*q^119 + 3*q^143 - 5*q^167 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 60, Eqs. (26.64),(26.65),(26.66)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

CF. A124226, A132969.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jul 20 2015 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, (n+1)\2, 1 - x^(2*k-1), 1 + x * O(x^n)) * sum(k=1, sqrtint(n), 2 * (-1)^k * x^k^2, 1), n))};

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

Expansion of phi(-q) + 2 * psi(-q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
Expansion of q^(1/24) * eta(q)^3 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -3, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 48^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A085140.
G.f.: ( Sum_{k in Z} (-1)^k * x^k^2 ) / ( Product_{k>0} (1 + x^k) ).
G.f.: Product_{k>0} (1 - x^k) / (1 + x^k)^2.
a(n) = (-1)^n * A132969(n). a(n) = A124226(n) unless n=1.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Oct 14 2017

A067357 Number of self-conjugate partitions of 4n+1 into odd parts.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 8, 10, 10, 12, 14, 15, 18, 20, 22, 26, 29, 32, 36, 40, 44, 50, 56, 60, 68, 76, 82, 92, 101, 110, 122, 134, 146, 160, 176, 191, 210, 230, 248, 272, 296, 320, 350, 380, 410, 446, 484, 522, 566, 612, 660, 715, 772, 830, 896, 966, 1038, 1120

Offset: 0

Naohiro Nomoto, Feb 24 2002

Also number of partitions of n in which even parts are distinct and if k occurs then so does every positive even number less than k (Dean Hickerson). Absolute values of the terms of A053254. - Emeric Deutsch, Feb 10 2006
The number of self-conjugate partitions of n into odd parts is nonzero if and only if n = 4*k + 1 for some nonnegative integer k. - Michael Somos, Jul 25 2015
Also number of C3v plane partitions of n = 3*k + 1 with rank 1 ; equivalently number of self-conjugate integer partitions with (weight-length) = n. - Wouter Meeussen, May 23 2025

			a(5)=3 because we have [11,1,1,1,1,1,1,1,1,1,1], [9,3,3,1,1,1,1,1,1] and [5,5,5,3,3].
G.f. = 1 + x + 2*x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + ...

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 260, Article 512.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. S. Andersen, A Bijection Between Two Different Classes of Partitions Enumerated by p_nu(n), arXiv:2007.09794 [math.CO], 2020.
George E. Andrews, The Bhargava-Adiga Summation and Partitions, Journal of the Indian Mathematical Society, Volume 84, Issue 3-4, 2017.
George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, to appear in Annals of Combinatorics, 2017.

Cf. A000700, A000701.

Cf. A053253, A053254, A085140, A047993.

g:=sum(q^(k*(k+1))/product(1-q^(2*j+1),j=0..k),k=0..8): gser:=series(g,q=0,80): seq(coeff(gser,q,n),n=0..75); # Emeric Deutsch, Feb 10 2006

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / Product[ 1 - x^i, {i, 1, 2 k + 1, 2}], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* Michael Somos, Jul 25 2015 *)
Table[Length[Flatten[Table[Select[IntegerPartitions[w], (w-Length[#])== r && TransposePartition[#] == # &],{w,r,1+2r}],1]],{r,1,17}] (* Wouter Meeussen, May 24 2025 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 4*n+1) -1) \ 2, x^(k^2 + k) / prod(j=0, k, 1 - x^(2*j+1), 1 + x * O(x^(n - k^2 - k)))), n))}; /* Michael Somos, Jan 27 2008 */

/* Continued Fraction Expansion: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 + (-x)^(n-k+1)*(1 - (-x)^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 09 2013

G.f.: Sum_{k>=0} q^(k*(k+1)) / ((1-q) * (1-q^3) ... (1-q^(2*k+1))). - Emeric Deutsch and Dean Hickerson
G.f.: Sum_{k>=0} q^k * (1+q) * (1+q^3) ... (1+q^(2*k-1)). - Dean Hickerson and Vladeta Jovovic
G.f.: 1/(1 - x*(1+x)/(1 + x^2*(1-x^2)/(1 - x^3*(1+x^3)/(1 + x^4*(1-x^4)/(1 - x^5*(1+x^5)/(1 - ...)))))), a continued fraction. - Paul D. Hanna, Jul 09 2013
From Michael Somos, Jul 25 2015: (Start)
Expansion of nu(-x) in powers of x where nu() is a 3rd-order mock theta function.
a(n) = (-1)^n * A053254(n).
a(2*n) = A085140(n).
a(2*n + 1) = A053253(n). (End)
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jun 15 2019

More terms from Emeric Deutsch, Feb 10 2006

A260574 Expansion of phi(x^3) * psi(x^3) / f(x) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 2, 0, 2, -1, 4, -2, 5, -2, 6, -2, 10, -4, 12, -4, 15, -6, 20, -8, 26, -9, 32, -12, 40, -16, 50, -18, 60, -22, 76, -28, 92, -33, 110, -40, 134, -50, 160, -58, 191, -70, 230, -84, 272, -98, 320, -116, 380, -138, 446, -160, 522, -188, 612, -222, 715, -256

Offset: 0

Michael Somos, Jul 29 2015

sign

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

			G.f. = 1 - x + 2*x^2 + 2*x^4 - x^5 + 4*x^6 - 2*x^7 + 5*x^8 - 2*x^9 + ...
G.f. = q - q^4 + 2*q^7 + 2*q^13 - q^16 + 4*q^19 - 2*q^22 + 5*q^25 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A085140, A097196, A257655.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3]^3 / (QPochhammer[ -x] EllipticTheta[ 4, 0, x^6]), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^3 * eta(x^12 + A)^2), n))};

q='q+O('q^99); Vec(eta(q)*eta(q^4)*eta(q^6)^7/(eta(q^2)^3*eta(q^3)^3*eta(q^12)^2)) \\ Altug Alkan, Aug 01 2018

Expansion of f(x^3)^3 / (f(x) * phi(-x^6)) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^3 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -1, 2, 2, 1, -1, -2, -1, 1, 2, 2, -1, -1, ...].
a(2*n) = A085140(n). a(2*n + 1) = - A097196(n). a(4*n + 1) = - A257655(n).

A029863 Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....

Original entry on oeis.org

1, 2, 6, 12, 27, 50, 98, 172, 310, 522, 888, 1444, 2357, 3724, 5882, 9072, 13957, 21082, 31732, 47072, 69545, 101540, 147620, 212516, 304631, 433054, 613030, 861616, 1206089, 1677766, 2324844, 3203748, 4398602, 6009390, 8181250

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n where there are 2 kinds of odd parts and 3 kinds of even parts. - Ilya Gutkovskiy, Jan 17 2018

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 27*x^4 + 50*x^5 + 98*x^6 + 172*x^7 + ...

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016, p. 16.
Songxin Liang & Jingzhong Zhang, A complete discrimination system for polynomials with complex coefficients and its automatic generation, 1999, p. 2. the sequence Cn is the numbers of cases for root classification of an n-degree polynomial with complex coefficients.
N. J. A. Sloane, Transforms

Cf. A002513, A010693, A085140, A262380.

nmax = 50; CoefficientList[Series[Product[1/((1 + x^k)*(1 - x^k)^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 20 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (eta(x + A)^2 * eta(x^2 + A)), n))};

Euler transform of period 2 sequence [2, 3, ...].
a(n) ~ 5 * exp(sqrt(5*n/3)*Pi) / (48 * n^(3/2)). - Vaclav Kotesovec, Sep 20 2015
G.f.: Product_{k >= 1} 1/(1-x^k)^A010693(k-1). - Georg Fischer, Dec 10 2020

A261998 Expansion of Product_{k>=1} (1-x^k)*(1+x^k)^4.

Original entry on oeis.org

1, 3, 5, 10, 17, 26, 43, 65, 95, 140, 201, 283, 395, 545, 740, 1002, 1343, 1780, 2350, 3077, 4002, 5183, 6670, 8535, 10880, 13801, 17426, 21925, 27475, 34297, 42677, 52926, 65415, 80625, 99077, 121403, 148386, 180890, 219960, 266857, 323002, 390086, 470125

Offset: 0

Vaclav Kotesovec, Sep 08 2015

nonn

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

Equals A000009 convolved with A085140. - George Beck, Jul 03 2016

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. 16.

Cf. A085140, A262380.

nmax = 80; CoefficientList[Series[Product[(1 - x^k) * (1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(5/2) * sqrt(n)).

cp's OEIS Frontend

A053254 Coefficients of the '3rd-order' mock theta function nu(q).

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A132970 Expansion of phi(-x) * chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

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A067357 Number of self-conjugate partitions of 4n+1 into odd parts.

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A260574 Expansion of phi(x^3) * psi(x^3) / f(x) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

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A029863 Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....

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A261998 Expansion of Product_{k>=1} (1-x^k)*(1+x^k)^4.

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