A328548 -id:A328548 - cp's OEIS frontend

A022567 Expansion of Product_{m>=1} (1+x^m)^2.

1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 238, 319, 426, 562, 736, 960, 1242, 1598, 2048, 2608, 3306, 4175, 5248, 6570, 8198, 10190, 12622, 15589, 19190, 23552, 28830, 35190, 42842, 52034, 63040, 76198, 91904, 110604, 132832, 159216, 190464, 227417

Offset: 0

N. J. A. Sloane, Jun 14 1998

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of n into distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*, thus a(4)=9. - Jon Perry, Apr 04 2004
Number of partitions of n into odd parts, each part being of two kinds. E.g., a(3)=6 because we have 3, 3', 1+1+1, 1+1+1', 1+1'+1', 1'+1'+1'. - Emeric Deutsch, Mar 22 2005
Euler transform of period 2 sequence [2,0,2,0,...]. - Emeric Deutsch, Mar 22 2005
Equals A000041 convolved with A010054. - Gary W. Adamson, Jun 11 2009
The sum of the least gaps in all partitions of n. The "least gap" of a partition is the least positive integer that is not a part of the partition. Example: a(4) = 9 because the least gaps in [4], [3,1], [2,2], [2,1,1], and [1,1,1,1] are 1, 2, 1, 3, and 2, respectively. - Emeric Deutsch, May 18 2015
Number of 2-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The least gap is also known as the minimal excludant or mex; see Andrews and Newman. - George Beck, Dec 10 2020

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 14*x^5 + 22*x^6 + 32*x^7 + 46*x^8 + ...
G.f. = q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + ...

P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Cristina Ballantine, Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Helena Bergold, Lukas Egeling, and Hung. P. Hoang, Signotopes with few plus signs, arXiv:2411.19208 [math.CO], 2024. See p. 14.
J. Currie, N. Rampersad, Binary words avoiding xx^Rx and strongly unimodal sequences, JIS 18 (2015) #15.10.3.
Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, arXiv preprint arXiv:1110.5103 [math.CO], 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 852
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.
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See q(n)'.
Mbavhalelo Mulokwe and Konstantinos Zoubos, Free fermions, neutrality and modular transformations, arXiv:2403.08531 [hep-th], 2024.
Michael Somos, Introduction to Ramanujan theta functions
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A022597, A285221, A304048.
Cf. A010054. - Gary W. Adamson, Jun 11 2009
Column k=2 of A286335.
Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Coefficients(&*[(1+x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

A022567 := proc(n)
    local x,m;
    product((1+x^m)^2,m=1..n) ;
    expand(%) ;
    coeff(%,x,n) ;
end proc: # R. J. Mathar, Jun 18 2016

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
(QPochhammer[-1, x]^2/4 + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j+1]] += poly[[j-k+1]], {j, nmax, k, -1}]; , {p, 1, 2}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 14 2017 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^2, n))}; /* Michael Somos, Mar 21 2004 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^2, n))}; /* Michael Somos, Jun 03 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 0, 2)
a = EulerTransform(b)
print([a(n) for n in range(45)]) # Peter Luschny, Nov 11 2020

a(n) = p(n)+p(n-1)+p(n-3)+p(n-6)+...+p(n-k*(k+1)/2)+..., where p() is A000041(). E.g. a(8) = p(8)+p(7)+p(5)+p(2) = 22+15+7+2 = 46. - Vladeta Jovovic, Aug 09 2004
Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q. - Michael Somos, Apr 27 2008
Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Apr 27 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022597. - Michael Somos, Apr 27 2008
G.f.: Product_{k>0} (1 + x^k)^2.
Convolution square of A000009. Convolution inverse of A022597. - Michael Somos, Apr 27 2008
Parity result: a(n) is even except when n is twice a generalized pentagonal number (i.e., of the form 2*A001318(m) for some m). - Peter Bala, Mar 19 2009
a(n) ~ exp(Pi * sqrt(2*n/3)) / (4 * 6^(1/4) * n^(3/4)) * (1 + (Pi/(12*sqrt(6)) - 3*sqrt(3/2)/(8*Pi)) / sqrt(n) + (Pi^2/1728 - 45/(256*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Mar 05 2015, extended Jan 22 2017
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

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

A263002 Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 5, 10, 20, 34, 61, 100, 165, 260, 408, 620, 940, 1390, 2045, 2960, 4257, 6040, 8525, 11900, 16522, 22738, 31130, 42300, 57210, 76872, 102834, 136800, 181230, 238900, 313725, 410160, 534330, 693330, 896655, 1155420, 1484274, 1900420, 2426215, 3088100

Offset: 0

Michael Somos, Oct 07 2015

nonn

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

Number of 5-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 34*x^5 + 61*x^6 + 100*x^7 + ...
G.f. = q + 2*q^4 + 5*q^7 + 10*q^10 + 20*q^13 + 34*q^16 + 61*q^19 + 100*q^22 + ...

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

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

Cf. A058511.

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

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(5); # N. J. A. Sloane, Oct 20 2019

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5] / QPochhammer[ x])^2, {x, 0, n}];

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

Expansion of q^(-1/3) * (eta(q^5) / eta(q))^2 in powers of q.
Euler transform of period 5 sequence [ 2, 2, 2, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (45 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A058511.
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (v - u^2) - 4*u^2*v^2.
Convolution inverse is A058511.
a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 3^(1/4) * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
See Maple code for a simple g.f. - N. J. A. Sloane, Oct 20 2019

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A022567 Expansion of Product_{m>=1} (1+x^m)^2.

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

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A263002 Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.

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A328547 Number of 3-regular bipartitions of n.

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