A001934 -id:A001934 - cp's OEIS frontend

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, 105, 132, 166, 208, 258, 320, 395, 484, 592, 722, 876, 1060, 1280, 1539, 1846, 2210, 2636, 3138, 3728, 4416, 5222, 6163, 7256, 8528, 10006, 11716, 13696, 15986, 18624, 21666, 25169, 29190, 33808, 39104, 45164

Offset: 0

N. J. A. Sloane, Simon Plouffe, Robert G. Wilson v

Also number of partitions of n where no part appears more than three times.
a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al.).
Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - Emeric Deutsch, Apr 19 2006
Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - Gary W. Adamson, Mar 16 2010
Equals left border of triangle A174715. - Gary W. Adamson, Mar 27 2010
The Cayley reference is actually to A083365. - Michael Somos, Feb 24 2011
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A000009 and A035457. - Vaclav Kotesovec, Aug 23 2015
Convolution inverse is A082303. - Michael Somos, Sep 30 2017
The g.f. in the form Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1-x^k) = Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1+x^k-2*x^k) == Sum_{n >= 0} x^(n*(n+1)/2) (mod 2). It follows that a(n) is odd iff n = k*(k + 1)/2 for some nonnegative integer k. Cf. A333374. - Peter Bala, Jan 08 2025

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 16*x^8 + 22*x^9 + ...
G.f. = q + q^9 + 2*q^17 + 3*q^25 + 4*q^33 + 6*q^41 + 9*q^49 + 12*q^57 + 16*q^65 + 22*q^73 + ...
a(5)=6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1].

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.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.2).
M. D. Hirschhorn, The Power of q, Springer, 2017. See ped page 303ff.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241.
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)
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 9.)
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See Theorem 1.4 p. 75.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Cristina Ballantine and Mircea Merca, Parity of sums of partition numbers and squares in arithmetic progressions, The Ramanujan Journal, 2016.
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.]
S.-C. Chen, On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943.
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
Joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?
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. 15.
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Alexander Patkowski, On some partitions where even parts do not repeat, Demonstratio Mathematica Volume 42, Issue 2 (Jun 2009), pp. 259-263.
Eric Weisstein's World of Mathematics, Partition Function b_k and Partition Function P.
Wikipedia, Glaisher's Theorem.

Cf. A000009, A000726, A001936, A035959, A035985, A042968, A061198, A061199, A070048, A081055, A081056, A083365, A098491, A098492, A219601.
Cf. A000041, A010054. - Gary W. Adamson, Mar 16 2010
Cf. A174715. - Gary W. Adamson, Mar 27 2010
Cf. A082303.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a001935 = p a042968_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Sep 02 2012

g:=product((1+x^j)*(1+x^(2*j)),j=1..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..48); # Emeric Deutsch, Apr 19 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 24 2015

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8), {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
CoefficientList[Series[Product[1+x^j+x^(2j)+x^(3j), {j,1,48}], {x,0,48}],x] (* Jean-François Alcover, May 26 2011, after Jon Perry *)
QP = QPochhammer; CoefficientList[QP[q^4]/QP[q] + O[q]^50, q] (* Jean-François Alcover, Nov 24 2015 *)
a[0] = 1; a[n_] := a[n] = Sum[a[n-j] DivisorSum[j, If[Divisible[#, 4], 0, #]&], {j, 1, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 4], 0, 2] ], {n, 0, 49}] (* Robert Price, Jul 28 2020 *)

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

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 8*n + 1) - 1)\2, prod(i=1, k, (1 + x^i) / (x^-i - 1), 1 + x * O(x^n))), n))}; /* Michael Somos, Jun 01 2004 */

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(1+(-x)^m+x*O(x^n))/m)),n)} \\ Paul D. Hanna, Jul 24 2013

Euler transform of period 4 sequence [ 1, 1, 1, 0, ...].
Expansion of q^(-1/8) * eta(q^4) / eta(q) in powers of q. - Michael Somos, Mar 19 2004
Expansion of psi(-x) / phi(-x) = psi(x) / phi(-x^2) = psi(x^2) / psi(-x) = chi(x) / chi(-x^2)^2 = 1 / (chi(x) * chi(-x)^2) = 1 / (chi(-x) * chi(-x^2)) = f(-x^4) / f(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 08 2011
G.f.: Product(j>=1, 1 + x^j + x^(2*j) + x^(3*j)). - Jon Perry, Mar 30 2004
G.f.: Product_{k>=1} (1+x^k)^(2-k%2). - Jon Perry, May 05 2005
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)) = 1 + Sum_{k>0}(Product_{i=1..k} (x^i + 1) / (x^-i - 1)).
G.f.: Sum_{n>=0} ( x^(n*(n+1)/2) * Product_{k=1..n} (1+x^k)/(1-x^k) ). - Joerg Arndt, Apr 07 2011
G.f.: P(x^4)/P(x) where P(x) = Product_{k>=1} 1-x^k. - Joerg Arndt, Jun 21 2011
A083365(n) = (-1)^n a(n). Convolution square is A001936. a(n) = A098491(n) + A098492(n). a(2*n) = A081055(n). a(2*n + 1) = A081056(n).
G.f.: (1+ 1/G(0))/2, where G(k) = 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
G.f.: exp( Sum_{n>=1} (x^n/n) / (1 + (-x)^n) ). - Paul D. Hanna, Jul 24 2013
a(n) ~ Pi * BesselI(1, sqrt(8*n + 1)*Pi/4) / (2*sqrt(8*n + 1)) ~ exp(Pi*sqrt(n/2)) / (4 * (2*n)^(3/4)) * (1 + (Pi/(16*sqrt(2)) - 3/(4*Pi*sqrt(2))) / sqrt(n) + (Pi^2/1024 - 15/(64*Pi^2) - 15/128) / n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082303. - Michael Somos, Sep 30 2017

More terms from James Sellers

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

Original entry on oeis.org

1, 2, 5, 10, 18, 32, 55, 90, 144, 226, 346, 522, 777, 1138, 1648, 2362, 3348, 4704, 6554, 9056, 12425, 16932, 22922, 30848, 41282, 54946, 72768, 95914, 125842, 164402, 213901, 277204, 357904, 460448, 590330, 754368, 960948, 1220370, 1545306

Offset: 0

N. J. A. Sloane

The Cayley reference is actually to A079006. - Michael Somos, Feb 24 2011
In the math overflow link is a conjecture that a(n) == a(9*n + 2) (mod 4).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of 4-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The g.f. in the form A(x) = Sum_{k >= 0} x^(k*(k+1)) / (1 + 2*Sum_{k >= 1} (-1)^k * x^(k^2)) == Sum_{k >= 0} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + 90*x^7 + 144*x^8 + ...
G.f. = q + 2*q^5 + 5*q^9 + 10*q^13 + 18*q^17 + 32*q^21 + 55*q^25 + 90*q^29 + ...

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. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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]
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
joro on Math Overflow, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?.
T. Kathiravan and S. N. Fathima, On L-regular bipartitions modulo L, The Ramanujan Journal 44.3 (2017): 549-558.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 07, 1976
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935, A022568, A079006, A082304, A098613, A127391, A127392, A210656.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548, A333374.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> [2,2,2,0] [modp(n-1,4)+1]): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
f:=(k,M) -> mul(1-q^(k*j),j=1..M); LRBP := (L,M) -> (f(L,M)/f(1,M))^2; S := L -> seriestolist(series(LRBP(L,80),q,60)); S(4); # N. J. A. Sloane, Oct 20 2019

m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* Jean-François Alcover, Sep 02 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]) / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^2, x^2])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)

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

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 / if(k%4, 1 - x^k, 1), 1 + x * O(x^n))^2, n))};

G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....
Convolution square of A001935. A079006(n) = (-1)^n a(n).
Expansion of q^(-1/4) * (1/2) * (k / k')^(1/2) in powers of q.
Euler transform of period 4 sequence [ 2, 2, 2, 0, ...].
Given g.f. A(x), then B(q) = (q * A(q^4))^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + 16*u) * (1 + 16*v) * v - u^2. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (1 + 4*u*v)^2. - Michael Somos, Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - x^k))^2.
Equals A000009 convolved with A098613. - Gary W. Adamson, Mar 24 2011
a(9*n + 2) = a(n) + 4 * A210656(3*n). - Michael Somos, Apr 02 2012
Convolution inverse is A082304. - Michael Somos, May 16 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082304. - Michael Somos, May 16 2015
Expansion of f(-x^4)^2 / f(-x)^2 = psi(x^2) / phi(-x) = psi(-x)^2 / phi(-x)^2 = psi(x)^2 / phi(-x^2)^2 = psi(x^2)^2 / psi(-x)^2 = chi(x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x)^4) = 1 / (chi(-x)^2 * chi(-x^2)^2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 16 2015
a(n) ~ exp(Pi*sqrt(n)) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 18 2015
G.f.: A(x) = Sum_{n >= 0} x^(n*(n+1)) / Sum_{n = -oo..oo} (-1)^n*x^(n^2). - Peter Bala, Feb 19 2021

A079006 Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.

Original entry on oeis.org

1, -2, 5, -10, 18, -32, 55, -90, 144, -226, 346, -522, 777, -1138, 1648, -2362, 3348, -4704, 6554, -9056, 12425, -16932, 22922, -30848, 41282, -54946, 72768, -95914, 125842, -164402, 213901, -277204, 357904, -460448, 590330, -754368, 960948, -1220370

Offset: 0

Michael Somos, Dec 22 2002

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

The Lagrange series reversion of Sum_{n >= 1} a(n-1)*x^n is Sum_{n >= 1} A002103(n-1)*x^n. See the example in A002103. - Wolfdieter Lang, Jul 09 2016

			G.f. A(x) = 1 - 2*x + 5*x^2 - 10*x^3 + 18*x^4 - 32*x^5 + 55*x^6 - 90*x^7 + 144*x^8 + ...
G.f. B(q) = q * A(q^4) = q - 2*q^5 + 5*q^9 - 10*q^13 + 18*q^17 - 32*q^21 + 55*q^25 - 90*q^29 + ...

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. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 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]
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012, eq. (8), p. 11 with eq. (2), p. 10.
Nicco, Conjectured identity of the product of two theta functions, Math Stackexchange, Sep 09 2015.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000700, A001936, A002103, A008441, A029839, A046897, A083365, A127391, A127392, A189925.

a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^2, {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ (m / 16 / q)^(1/4), {q, 0, n}]]; (* Michael Somos, Jul 08 2011 *)
QP = QPochhammer; s = (QP[q]*(QP[q^4]^2/QP[q^2]^3))^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(2*k))^4 / (1+x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 / QPochhammer[ -x]^2, {x, 0, n}]; (* Michael Somos, Apr 19 2017 *)

{a(n) = my(N, A); if( n<0, 0, N = (sqrtint(16*n + 1) + 1)\2; A = contfracpnqn( matrix(2, N, i, j, if( i==1, if( j<2, 1 + O(x^(N^2 + N)), (x^(j-1) + x^(3*j - 3))^2), 1 - x^(4*j - 2)))); polcoeff( A[2,1] / A[1,1], 4*n))}; /* Michael Somos, Sep 01 2005 */

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m = 1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A / (1 + 4 * x*A^2))); polcoeff(A, 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)^3)^2, n))};

a(n) = (2/n)*Sum_{k=1..n} (-1)^k*A046897(k)*a(n-k). - Vladeta Jovovic, Dec 24 2002
Expansion of q^(-1/4) * (1/2) * k^(1/2) in powers of q, where k^2 is the parameter and q the Jacobi nome of elliptic functions.
Expansion of (1/(2*q)) * (1 - sqrt(k')) / (1 + sqrt(k')) in powers of q^4, where k'^2 is the complementary parameter and q the Jacobi nome of elliptic functions. See the Fricke reference.
Expansion of psi(x^2) / phi(x) = psi(x)^2 / phi(x)^2 = psi(x^2)^2 / psi(x)^2 = psi(-x)^2 / phi(-x^2)^2 = chi(-x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x^2)^2) = 1 / (chi(x)^4 * chi(-x)^2) = f(-x^4)^2 / f(x)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [-2, 4, -2, 0, ...].
G.f. A(x) satisfies A(x)^2 = A(x^2) / (1 + 4 * x * A(x^2)^2). - Michael Somos, Mar 19 2004
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 * (1 + 4 * v^2) - v. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u3 * (u6 + u2)^2 - u2*u6. - Michael Somos, Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - (-x)^k))^2.
Expansion of continued fraction 1 / (1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...))) in powers of x^4. - Michael Somos, Sep 01 2005
Given g.f. A(x), then B(q) = 2 * q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4 . - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A189925.
Convolution inverse of A029839. Convolution square of A083365. a(n) = (-1)^n * A001936(n).
G.f.: 1/Q(0), where Q(k)= 1 - x^(k+1/2) + (x^((k+1)/4) + x^((3*k+3)/4))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(7/2)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
Given g.f. A(x), and B(x) is the g.f. for A008441, then A(x) = B(x^2) / B(x) and A(x) * A(x^2) * A(x^4) * ... = 1 / B(x). - Michael Somos, Apr 20 2017
Expansion of continued fraction 1 / (1 - x^1 + x^1*(1 + x^1)^2 / (1 - x^3 + x^2*(1 + x^2)^2 / (1 - x^5 + x^3*(1 + x^3)^2 / ...))) in powers of x^2. - Michael Somos, Apr 20 2017
a(n) = A208933(4*n+1) - A215348(4*n+1) (conjectured). - Thomas Baruchel, May 14 2018
A(x^4) = (1/(m*x)) * ( chi(x)^m - chi(-x)^m ) / ( chi(x)^m + chi(-x)^m ) at m = 2, where chi(x)  = Product_{i >= 0} (1 + x^(2*i+1)) is the g.f. of A000700. The formula gives generating functions related to A092869 when m = 1 and A001938 (also A093160) when m = 4. - Peter Bala, Sep 23 2023

A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).

Original entry on oeis.org

1, -4, 14, -40, 101, -236, 518, -1080, 2162, -4180, 7840, -14328, 25591, -44776, 76918, -129952, 216240, -354864, 574958, -920600, 1457946, -2285452, 3548550, -5460592, 8332425, -12614088, 18953310, -28276968, 41904208, -61702876, 90304598, -131399624

Offset: 0

N. J. A. Sloane

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

k^2 is the parameter and q the Jacobi nome of elliptic functions. See, e.g., Fricke, p. 11, eq. (8) with p. 10. eq. (1). - Wolfdieter Lang, Jul 04 2016

			G.f. = 1 - 4*x + 14*x^2 - 40*x^3 + 101*x^4 - 236*x^5 + 518*x^6 - 1080*x^7 + ...
G.f. of B(q) = q * A(q^2): q - 4*q^3 + 14*q^5 - 40*q^7 + 101*q^9 - 236*q^11 + 518*q^13 - 1080*q^15 + ...

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.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 385.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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]
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000122, A000700, A001936, A010054, A079006, A093160, A121373, A127393, A127931, A127932, A139820.

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - (-x)^k, {k, n}])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)] / (2 EllipticTheta[ 3, 0, q]))^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)

{a(n) = my(A, A2, m); if( n<0, 0, n = 2*n + 1; A = x + O(x^3); m=2; while( mMichael Somos, Mar 26 2004 */

{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)^4, n))}; /* Michael Somos, Mar 26 2004 */

Expansion of (psi(x^2) / phi(x))^2 = (psi(x) / phi(x))^4 = (psi(x^2) / psi(x))^4 = (psi(-x) / psi(-x^2))^4 = (chi(-x) / chi(-x^2)^2)^4 = (chi(x)^2 * chi(-x))^-4 = (chi(x) * chi(-x^2))^-4 = (f(-x^4) / f(x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Feb 26 2012
G.f. A(x) satisfies 1 = (1 - 16 * x * A(x)^2) * (1 + 16 * x * A(-x)^2). - Michael Somos, Mar 26 2004
Expansion of q^(-1/2) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^4 in powers of q.
Euler transform of period 4 sequence [ -4, 8, -4, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v - (u * (1 + 4*v))^2. - Michael Somos, Mar 26 2004
G.f. A(q) satisfies A(q) = sqrt(A(q^2)) / (1 + 4*q*A(q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. - Joerg Arndt, Aug 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^4.
a(n) = (-1)^n * A093160(n). Convolution square of A079006.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139820. - Michael Somos, Jun 04 2015
G.f.: ((Sum_{n >= 0} x^(n*(n+1))) / (1 + Sum_{n >= 1} x^(n^2)))^4 (from the sum representation of the Jacobi theta functions evaluated at vanishing argument). - Wolfdieter Lang, Jul 04 2016
a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Edited by N. J. A. Sloane, Mar 31 2007

A002513 Number of "cubic partitions" of n: expansion of Product_{k>0} 1/((1-x^(2k))^2*(1-x^(2k-1))) in powers of x.

Original entry on oeis.org

1, 1, 3, 4, 9, 12, 23, 31, 54, 73, 118, 159, 246, 329, 489, 651, 940, 1242, 1751, 2298, 3177, 4142, 5630, 7293, 9776, 12584, 16659, 21320, 27922, 35532, 46092, 58342, 75039, 94503, 120615, 151173, 191611, 239060, 301086, 374026, 468342, 579408, 721638, 889287

Offset: 0

N. J. A. Sloane, Simon Plouffe

For a real polynomial equation of degree n, a(n) is the number of possibilities for the roots to be real and unequal, real and equal (in various combinations), or simple or multiple complex conjugates. For example, a(3)=4 because we can have: three equal roots, two equal roots, three distinct real roots and two complex roots (see the Monthly Problem reference). - Emeric Deutsch, Mar 22 2005
Number of partitions of n, the even parts being of two kinds. E.g. a(4)=9 because we have 4, 4', 3+1, 2+2, 2+2', 2'+2', 2+1+1, 2'+1+1, 1+1+1+1. - Emeric Deutsch, Mar 22 2005
For the name "cubic partition" see Xiong; Chen & Lin; Chern & Dastidar. - Michel Marcus, Jan 28 2016

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 23*x^6 + 31*x^7 + 54*x^8 + 73*x^9 + ...
G.f. = 1/q + q^7 + 3*q^15 + 4*q^23 + 9*q^31 + 12*q^39 + 23*q^47 + 31*q^55 + 54*q^63 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0930 and N0931).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Zakir Ahmed, Nayandeep Deka Baruah, and Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Koustav Banerjee, New Asymptotics and Inequalities Related to the Partition Function, Doctoral Thesis, Johannes Kepler Univ. (Linz, Austria 2022).
M. F. Capobianco and C. F. Pinzka, Problem 2055, Amer. Math. Monthly, 75 (1968), 188; 76 (1969), 194.
William Y. C. Chen and Bernard L. S. Lin, Congruences for the Number of Cubic Partitions Derived from Modular Forms, arXiv:0910.1263 [math.NT], 2016.
Shane Chern and Manosij Ghosh Dastidar, Congruences and recursions for the cubic partitions, arXiv:1601.06480 [math.NT], 2016.
Marston Conder, Tomaš Pisanski, and Arjana Žitnik, Vertex-transitive graphs and their arc-types, arXiv preprint arXiv:1505.02029 [math.CO], 2015.
R. K. Guy, Letter to Morris Newman, Aug 21 1986, concerning A2513 (annotated scanned copy, with permission).
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024. Table 1 p. 5 mentions this sequence.
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.
Vaclav Kotesovec, Asymptotics of sequence A002513, 2019.
Lukas Mauth, Exact formula for cubic partitions, arXiv:2305.03396 [math.NT], 2023.
Morris Newman, Construction and application of a class of modular functions (II). Proc. London Math. Soc. (3) 9 1959 373-387.
Morris Newman, Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]
James A. Sellers, Elementary proofs of congruences for the cubic and overcubic partition functions, Australasian Journal of Combinatorics, Volume 60(2) (2014), Pages 191-197.
Xinhua Xiong, The number of cubic partitions modulo powers of 5, arXiv:1004.4737 [math.NT], 2010.

Cf. A000122, A015128, A001934, A089799, A215947, A273225, A319455.

N:= 50: # to get a(0) to a(N)
P:= mul((1-x^(2*k))^(-2)*(1-x^(2*k-1))^(-1),k=1..ceil(N/2)):
S:= series(P, x, N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 26 2016
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(d::odd, d, 2*d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 05 2020

max = 50; f[x_] := Product[ 1/((1-x^(2 k))^2*(1-x^(2k-1))), {k, 1, Ceiling[max/2]} ]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 04 2011 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q] / QPochhammer[ q^2], {q, 0, n}];(* Michael Somos, Jul 17 2013 *)
Table[Sum[PartitionsP[k]*PartitionsP[n-2k],{k,0,n/2}],{n,0,50}] (* Vaclav Kotesovec, Jun 22 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / eta(x + A) / eta(x^2 + A), n))}; /* Michael Somos, Nov 10 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 2)
a = EulerTransform(b)
print([a(n) for n in range(44)]) # Peter Luschny, Nov 17 2022

From Michael Somos, Mar 23 2003: (Start)
Expansion of q^(1/8) / (eta(q) * eta(q^2)) in powers of q.
Euler transform of period 2 sequence [1, 2, ...].
G.f.: Product_{k>0} 1/((1 - x^(2*k))^2 * (1 - x^(2*k-1))).
(End)
Given g.f. A(x), then B(q) = A(q)^8 / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 16*v^4 + v^3*w + 256*u*v^3 + 16*u*v^2*w - u^2*w^2. - Michael Somos, Apr 03 2005
a(n) ~ exp(Pi*sqrt(n)) / (8*n^(5/4)) * (1 - (Pi/16 + 15/(8*Pi))/sqrt(n)). - Vaclav Kotesovec, Jun 22 2015, extended Jan 17 2017
From Michel Marcus, Jan 28 2016: (Start)
G.f.: Product_{k>0} 1/((1 - x^k) * (1 - x^(2*k))).
a(3n+2) = 0 (mod 3).
a(25n+22) = 0 (mod 5) (see Xiong).
a(49n+15) = a(49n+29) = a(49n+36) = a(49n+43) = 0 (mod 7) (see Chen & Lin).
a(297n+62) = a(297n+161) = 0 (mod 11) (see Chern & Dastidar).
(End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(-7/2) (t/i)^-1 f(t) where q = exp(2 Pi i t). - Michael Somos, Oct 17 2017
G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018
From Peter Bala, Sep 25 2023: (Start)
The g.f. A(x) satisfies log(A(x)) = x + 5*x^2/2 + 4*x^3/3 + 13*x^4/4 + ... = Sum_{n >= 1} A215947(n)*x^n/n.
A(x^2) = 4/(F(x)*F(-x)) = 2/(F(x)*G(-x)), where F(x) = Sum_{n = -oo..oo} x^(n*(n+1)/2) is the g.f. of A089799 and G(x) = Sum_{n = -oo..oo} x^(n^2) is the g.f. of A000122. Cf. A001934. Note that 4/(F(-x)*F(-x)) is the g.f. of A273225.
The self-convolution A(x)^2 is the g.f. of A319455. (End)

More terms and information from Michael Somos, Mar 23 2003

A083365 Expansion of psi(x) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 2, -3, 4, -6, 9, -12, 16, -22, 29, -38, 50, -64, 82, -105, 132, -166, 208, -258, 320, -395, 484, -592, 722, -876, 1060, -1280, 1539, -1846, 2210, -2636, 3138, -3728, 4416, -5222, 6163, -7256, 8528, -10006, 11716, -13696, 15986, -18624, 21666, -25169, 29190, -33808, 39104

Offset: 0

Michael Somos, Apr 24 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A079006.
Convolution inverse is A029838.

			G.f. = 1 - x + 2*x^2 - 3*x^3 + 4*x^4 - 6*x^5 + 9*x^6 - 12*x^7 + 16*x^8 - 22*x^9 + ...
G.f. = q - q^9 + 2*q^17 - 3*q^25 + 4*q^33 - 6*q^41 + 9*q^49 - 12*q^57 + 16*q^65 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).
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.
H. T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York 1962, p. 170 MR0181773 (31 #6000)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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]
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.1),(9.3).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol

Cf. A001935, A029838, A108494.

(psi(x) / phi(x))^b: this sequence (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), A320049 (b=6), A320050 (b=7).

phi[x_] := EllipticTheta[3, 0, x]; psi[x_] := (1/2)*x^(-1/8)*EllipticTheta[2, 0, x^(1/2)]; s = Series[ psi[x]/phi[x], {x, 0, 100}]; A083365 = CoefficientList[s, x] (* Jean-François Alcover, Feb 18 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k))^2/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
(QPochhammer[-x^2, x^2, -1/2] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] / QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Oct 10 2019~ *)

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A / (1 + 4 * x * A^2))); polcoeff(sqrt(A), n))};

{a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1 + if( j>1, x^(j-1))))); polcoeff(A[2,1] / A[1,1] + x * O(x^n), 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)^3, n))};

Expansion of f(-x^4) / f(x) = psi(x) / phi(x) = psi(x^2) / psi(x) = psi(-x) / phi(-x^2) = 1 / (chi(x) * chi(-x^2)) = 1 / (chi^2(x) * chi(-x)) = chi(-x) / chi^2(-x^2) = (psi(x^2) / phi(x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of k^(1/4) / (2^(1/2) * q^(1/8)) in powers of q where k is elliptic modulus and q is the nome.
Expansion of q^(-1/8) * eta(q) * eta(q^4)^2 / eta(q^2)^3 in powers of q.
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^4 * (1 + 4*v^4).
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = v^4 - u^4 + u*v + 4*(u*v)^3.
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = w - u^2*v*(1 + 2*w^2). - Michael Somos, May 29 2005
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u6 - u1*u3 * (u2^2 + u6^2). - Michael Somos, May 29 2005
Given g.f. A(x), then B(q) = sqrt(2) * q * A(q^8) satisfies 0 = f(B(q), B(q^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
Euler transform of period 4 sequence [-1, 2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A108494. - Michael Somos, Feb 29 2012
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)) = (Sum_{k>0} x^(k^2 - k)) / (Sum_{k>0} x^((k^2 - k)/2)).
G.f.: 1 / (1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...)))).
A001935(n) = (-1)^n a(n).
G.f.: (1+1/Q(0))/2, where Q(k)= 1 + x^(k+1) + x^(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2))/(2^(11/4)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
G.f.: (-x^2; x^2){-1/2} = ((-1; x^2){1/2})/2, where (a; q)n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109506(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 14 2017
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(11/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 + x^k))). - Ilya Gutkovskiy, May 28 2018

A104794 Expansion of theta_4(q)^2 in powers of q.

Original entry on oeis.org

1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, 0, 0, -12, 8, 0, 0, -8, 0, 0, 4, 0, 8, 0, 4, -8, 0, 0, 8, -8, 0, 0, 0, -8, 0, 0, 0, -4, 12, 0, 8, -8, 0, 0, 0, 0, 8, 0, 0, -8, 0, 0, 4, -16, 0, 0, 8, 0, 0, 0, 4, -8, 8, 0, 0, 0, 0, 0, 8

Offset: 0

Michael Somos, Mar 26 2005

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Quadratic AGM theta functions: a(q) (see A004018), b(q) (A104794), c(q) (A005883).
In the Arithmetic-Geometric Mean, if a = theta_3(q)^2, b = theta_4(q)^2 then a' := (a+b)/2 = theta_3(q^2)^2, b' := sqrt(a*b) = theta_4(q^2)^2.

			G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000203, A000729, A001934, A002131, A004018, A008441, A053692, A113407, A113652, A122856, A122865, A204531, A227695, A246950, A256014, A258210.

# JacobiTheta4 is defined in A002448.
A104794List(len) = JacobiTheta4(len, 2)
A104794List(102) |> println # Peter Luschny, Mar 12 2018

A := Basis( ModularForms( Gamma1(8), 1), 100); A[1] - 4*A[2] + 4*A[3]; /* Michael Somos, Jan 31 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[1 - m] EllipticK[m] / (Pi/2), {q, 0, n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/4) EllipticK[m] / (Pi/2), {q, 0, 2 n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 4 DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jun 06 2015 *)

{a(n) = if( n<1, n==0, (-1)^n * 4 * sumdiv(n, d, (d%4==1) - (d%4==3)))};

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

{a(n) = if( n<0, 0, polcoeff( 1 + 4 * sum( k=1, n, (-x)^k / (1 + x^(2*k)), x * O(x^n)), n))};

Expansion of phi(-q)^2 = 2 * phi(q^2)^2 - phi(q)^2 = (phi(q) - 2*phi(q^4))^2 = f(-q)^3 / psi(q) = phi(-q^2)^4 / phi(q)^2 = psi(-q)^4 / psi(q^2)^2 = psi(q)^2 * chi(-q)^6 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (1-k^2)^(1/2) K(k^2) / (Pi/2) in powers of q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind.
Expansion of  K(k^2) / (Pi/2) in powers of -q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind. - Michael Somos, Jun 08 2015
Expansion of eta(q)^4 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -4, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u^2 + v^2) - 2*u*w^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 - 2*u1*u3 + 4*u2*u6 - 3*u3^2.
Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 16 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008441.
G.f.: theta_4(q)^2 = (Sum_{k in Z} (-q)^(k^2))^2 = (Product_{k>0} (1 - q^(2*k)) * (1 - q^(2*k - 1))^2)^2.
G.f.: 1 + 4 * Sum_{k>0} (-x)^k / (1 + x^(2*k)). - Michael Somos, Jun 08 2015
a(4*n + 3) = 0. a(n) = (-1)^n * A004018(n) = a(2*n). a(4*n + 1) = -4 * A008441(n). a(n) = -4 * A113652(n) unless n=0. a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = 4 * A122856(n). a(8*n + 1) = -4 * A113407(n). a(8*n + 5) = -8 * A053692(n).
a(n) = a(9*n) = A204531(8*n) = A246950(8*n) = A256014(9*n) = A258210(n). - Michael Somos, Jun 08 2015
Convolution inverse of A001934. Convolution with A000729 is A227695. - Michael Somos, Jun 08 2015
G.f.: 2 * Sum_{k in Z} (-1)^k * x^(k*(k + 1)/2) / (1 + x^k). - Michael Somos, Nov 05 2015
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017
G.f.: exp(2*Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 12, 30, 68, 144, 289, 556, 1034, 1868, 3292, 5678, 9608, 15984, 26188, 42314, 67509, 106460, 166090, 256552, 392628, 595696, 896484, 1338894, 1985298, 2923840, 4278448, 6222518, 8997544, 12938368, 18507297, 26340040, 37307326, 52597320, 73825504, 103180702

Offset: 2

Ilya Gutkovskiy, Jan 30 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A001934, A002448, A014968, A015128, A048574, A063725, A327380.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..37);  # Alois P. Heinz, Feb 10 2021

nmax = 37; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^2/4, {x, 0, nmax}], x] // Drop[#, 2] &
nmax = 37; CoefficientList[Series[(1/4) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
A015128[n_] := Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; a[n_] := (1/4) Sum[A015128[k] A015128[n - k], {k, 1, n - 1}]; Table[a[n], {n, 2, 37}]

G.f.: (1/4) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^2.
a(n) = Sum_{k=0..n} A014968(k) * A014968(n-k).
a(n) = (1/4) * Sum_{k=1..n-1} A015128(k) * A015128(n-k).
a(n) = (A001934(n) - 2 * A015128(n)) / 4 for n > 0.

A261519 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2^k).

Original entry on oeis.org

1, 4, 16, 60, 208, 692, 2224, 6940, 21152, 63188, 185488, 536268, 1529648, 4310804, 12017264, 33171916, 90745472, 246201412, 662897232, 1772295020, 4707336848, 12426673188, 32617079280, 85152717404, 221183486496, 571784014244, 1471463190032, 3770577250716

Offset: 0

Vaclav Kotesovec, Aug 23 2015

nonn

Convolution of A034899 and A102866.

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, A261520, A260916, A001934, A015128.

nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(2^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^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)*(2^(2*j)-1)) = 0.2545212486386431009939814261118792033...

A001937 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, 15593702, 21232521, 28790667, 38884082

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

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

The Cayley reference is actually to A187053. - Michael Somos, Jul 26 2012

			1 + 3*x + 9*x^2 + 22*x^3 + 48*x^4 + 99*x^5 + 194*x^6 + 363*x^7 + 657*x^8 + ...
q^3 + 3*q^11 + 9*q^19 + 22*q^27 + 48*q^35 + 99*q^43 + 194*q^51 + 363*q^59 + ...

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]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001936, A079006, A001935, A083365, A001938, A093160, A001939, A001940, A001941.

g100:= mul((1+x^(2*k))/(1-x^(2*k-1)),k=1..50)^3:
S:= series(g100,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Nov 30 2015

CoefficientList[ Series[Product[(1 - x^k)^(-3*Boole[Mod[k, 4] != 0]), {k, 1, 101}], {x, 0, 100}], x] (* Olivier GERARD, May 06 2009 *)
QP = QPochhammer; s = (QP[q^4]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 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))^3, n))} /* Michael Somos, Mar 06 2011 */

Expansion of q^(-3/8) * (eta(q^4) / eta(q))^3 in powers of q. - Michael Somos, Jul 26 2012
Euler transform of period 4 sequence [ 3, 3, 3, 0, ...]. - Michael Somos, Mar 06 2011
Convolution cube of A001935. A187053(n) = (-1)^n * a(n). - Michael Somos, Mar 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)))^3.
a(n) ~ 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Corrected and extended by Simon Plouffe

Checked and more terms from Olivier GERARD, May 06 2009

cp's OEIS Frontend

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

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A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

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A079006 Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.

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A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).

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A002513 Number of "cubic partitions" of n: expansion of Product_{k>0} 1/((1-x^(2k))^2*(1-x^(2k-1))) in powers of x.

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