A098151 -id:A098151 - cp's OEIS frontend

A261797 Expansion of Product_{k>=1} (1-x^(3k))(1-x^(5*k))/(1-x^k).

1, 1, 2, 2, 4, 4, 6, 7, 11, 12, 16, 19, 25, 29, 37, 43, 55, 63, 78, 90, 110, 127, 153, 176, 211, 242, 286, 328, 386, 441, 515, 586, 682, 775, 895, 1016, 1169, 1323, 1514, 1711, 1953, 2201, 2502, 2815, 3191, 3582, 4048, 4536, 5113, 5719, 6429, 7179, 8052

Offset: 0

Vaclav Kotesovec, Sep 01 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A103257, A098151, A138526, A261796.

nmax=80; CoefficientList[Series[Product[(1-x^(3*k))*(1-x^(5*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(14*n/5)/3) / sqrt(30*n).

A275633 Andrews's shadow difference function D_3(q).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 20, 31, 41, 56, 74, 101, 129, 172, 219, 284, 363, 464, 581, 738, 924, 1155, 1435, 1785, 2199, 2717, 3332, 4084, 4987, 6076, 7375, 8949, 10817, 13051, 15706, 18877, 22622, 27078, 32332, 38545, 45870, 54496, 64618, 76525, 90463, 106788, 125863, 148145, 174106

Offset: 0

N. J. A. Sloane, Aug 09 2016

nonn

Agrees with A237833 just for n <= 21.

George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Cf. A098151, A237833, A275632.

F:=(a,q,n)->mul(1-a*q^i,i=0..n-1); # This is (a;q)_n
M:=15;
# A098151:
THETA3:=(add((-1)^n*q^(3*n^2),n=-M..M)) /(add((-1)^n*q^(n^2),n=-M..M));
s1:=series(THETA3,q,80); seriestolist(%);
# A275632:
THETABAR3:=1+2*add( (F(q,q,n-1)*q^(n^2)) / (F(q^n,q,n)*(1-q^n)), n=1..M);
s2:=series(THETABAR3,q,80); seriestolist(%);
# A275633:
series((s1-s2)/8,q,80); seriestolist(%);

Equals (A098151-A275632)/8.

A100823 G.f.: Product_{k>0} (1+x^k)/((1-x^k)(1+x^(3k))(1+x^(5k))).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 69, 101, 146, 208, 293, 408, 563, 768, 1040, 1397, 1864, 2470, 3254, 4261, 5550, 7192, 9277, 11911, 15229, 19391, 24597, 31085, 39150, 49142, 61489, 76702, 95401, 118324, 146362, 180573, 222226, 272826, 334173, 408394, 498022

Offset: 0

Noureddine Chair, Jan 06 2005

nonn

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

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 46*x^7 + ...
G.f. = q^-1 + 2*q^2 + 4*q^5 + 7*q^8 + 12*q^11 + 19*q^14 + 30*q^17 + 46*q^20 + ...

Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2001 terms from Vaclav Kotesovec)
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. 17.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035939, A098151, A102346, A122129.

series(product((1+x^k)/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))),k=1..100),x=0,100);

CoefficientList[ Series[ Product[(1 + x^k)/((1 - x^k)*(1 + x^(3k))*(1 + x^(5k))), {k, 100}], {x, 0, 45}], x] (* Robert G. Wilson v, Jan 12 2005 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(5*k-1))*(1+x^(5*k-2))*(1+x^(5*k-3))*(1+x^(5*k-4)) / ((1-x^(6*k))*(1-x^(3*k-1))*(1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] QPochhammer[ x^5, x^10] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Mar 07 2016 *)

q='q+O('q^33); E(k)=eta(q^k);
Vec( (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) ) \\ Joerg Arndt, Sep 01 2015

a(n) ~ exp(Pi*sqrt(37*n/5)/3) * sqrt(37) / (12*sqrt(5)*n). - Vaclav Kotesovec, Sep 01 2015
G.f.: (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) where E(k) = prod(n>=1, 1-q^k ). - Joerg Arndt, Sep 01 2015
Euler transform of period 30 sequence [ 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, ...]. - Michael Somos, Mar 07 2016
Expansion of chi(-x^3) * chi(-x^5) / phi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Mar 07 2016
a(n) - A035939(2*n + 1) = A122129(2*n + 1). - Michael Somos, Mar 07 2016

More terms from Robert G. Wilson v, Jan 12 2005
Offset corrected by Vaclav Kotesovec, Sep 01 2015
a(14) = 563 <- 562 corrected by Vaclav Kotesovec, Sep 01 2015

A101230 Number of partitions of 2n in which both odd parts and parts that are multiples of 3 occur with even multiplicities. There is no restriction on the other even parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 20, 32, 50, 76, 113, 166, 240, 343, 484, 676, 935, 1282, 1744, 2355, 3158, 4208, 5573, 7340, 9616, 12536, 16266, 21012, 27028, 34628, 44196, 56204, 71226, 89964, 113270, 142180, 177948, 222089, 276430, 343172, 424959, 524966

Offset: 0

Noureddine Chair, Dec 16 2004

nonn

Note that if a partition of n has odd parts occur with even multiplicities then n must be even. This is the reason for only looking at partitions of 2n. - Michael Somos, Mar 04 2012

			a(8)=12 because 8 = 4+4 = 4+2+2 = 4+2+1+1 = 4+1+1+1+1 = 3+3+2 = 3+3+1+1 = 2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1.
1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + 113*x^9 + ...
1/q + 2*q^7 + 4*q^15 + 7*q^23 + 12*q^31 + 20*q^39 + 32*q^47 + 50*q^55 + 76*q^63 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Cf. A000041, A003105, A015128, A089812, A098151.

series(product((1+x^k)/((1-x^k)*(1+x^(3*k))),k=1..100),x=0,100);

nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1))*(1+x^(3*k-2)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))} /* Michael Somos, Mar 04 2012 */

G.f.: product_{k>0}(1+x^k)/((1-x^k)(1+x^(3k)))= Theta_4(0, x^3)/theta(0, x)1/product_{k>0}(1-x^(3k)).
Euler transform of period 6 sequence [2, 1, 1, 1, 2, 1, ...]. - Vladeta Jovovic, Dec 17 2004
Expansion of q^(1/8) * eta(q^2) * eta(q^3) / (eta(q)^2 * eta(q^6)) in powers of q. - Michael Somos, Mar 04 2012
Convolution inverse of A089812. - Michael Somos, Mar 04 2012
Convolution product of A000041 and A003105. - Michael Somos, Mar 04 2012
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*n). - Vaclav Kotesovec, Sep 01 2015

A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 6, 8, 10, 10, 12, 16, 22, 28, 32, 36, 42, 52, 66, 80, 92, 104, 120, 144, 174, 206, 236, 266, 304, 356, 420, 488, 554, 624, 708, 816, 946, 1084, 1224, 1372, 1548, 1764, 2016, 2288, 2568, 2868, 3216, 3632, 4110, 4626, 5166, 5748, 6412, 7188

Offset: 0

Noureddine Chair, Feb 15 2005

Convolution of A098884 and A003105. [corrected by Vaclav Kotesovec, Feb 07 2021]

Also equal to the number of overpartitions of n into parts congruent to 1 or 5 modulo 6. - Jeremy Lovejoy, Nov 28 2024

			E.g. a(7)=8 because 14=10+4=10+2+1+1=8+4+2=8+4+1+1=7+7=5+5+4=5+5+2+1+1.

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A003105, A098884, A080054, A001082, A098151, A015128.

series(product(((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))),k=1..100),x=0,100);
# alternative program:
with(gfun): series( add(x^(n*(3*n-2)), n = -6..6)/add((-1)^n*x^(n*(3*n-2)), n = -6..6), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

nmax = 50; CoefficientList[Series[Product[((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

G.f.: (Theta_4(0, x^2)*theta_4(0, x^3))/(theta_4(0, x)*theta_4(0, x^(6))) = Product_{k>0}((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))).
Euler transform of period 12 sequence [2, -1, 0, 0, 2, 0, 2, 0, 0, -1, 2, 0, ...]. - Vladeta Jovovic, Feb 17 2005
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 01 2015
G.f.: f(x,x^5)/f(-x,-x^5) = ( Sum_{n = -oo..oo} x^(n*(3*n-2)) )/( Sum_{n = -oo..oo} (-1)^n*x^(n*(3*n-2)) ), where f(a,b) = Sum_{n = -oo..oo} a^(n*(n+1)/2)*b^(n*(n-1)/2) is Ramanujan's 2-variable theta function. Cf. A080054 and A098151. - Peter Bala, Feb 05 2021

Example corrected by Vaclav Kotesovec, Sep 01 2015

A252706 Expansion of phi(-q) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 2, -2, 0, 4, -4, 0, 6, -8, 0, 10, -12, 0, 16, -18, 0, 24, -28, 0, 36, -40, 0, 52, -58, 0, 74, -84, 0, 104, -116, 0, 144, -160, 0, 198, -220, 0, 268, -296, 0, 360, -396, 0, 480, -528, 0, 634, -694, 0, 832, -908, 0, 1084, -1184, 0, 1404, -1528, 0

Offset: 0

Michael Somos, Apr 04 2015

sign

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

			G.f. = 1 - 2*q + 2*q^3 - 2*q^4 + 4*q^6 - 4*q^7 + 6*q^9 - 8*q^10 + 10*q^12 + ...

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. A097197, A098151, A101195, A139137.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];

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

Expansion of f(-q, -q^2) / f(q, q^2) in powers of q where f(,) is Ramanujan's two-variable theta function.
Euler transform of period 6 sequence [ -2, -1, 0, -1, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A101195.
G.f.: Product_{k>0} (1 - x^k + x^(2*k)) / (1 + x^k + x^(2*k)).
a(n) = (-1)^n * A139137(n).
Convolution inverse is A098151.
a(3*n + 2) = 0. a(3*n) = A098151(n). a(3*n + 1) = -2 * A097197(n).

A275632 Andrews's 4-shadow function THETA.BAR_3(q).

Original entry on oeis.org

1, 2, 4, 6, 10, 8, 16, 12, 20, 18, 24, 16, 38, 20, 32, 32, 42, 24, 52, 28, 56, 44, 48, 32, 80, 42, 56, 54, 76, 40, 96, 44, 84, 64, 72, 64, 122, 52, 80, 76, 120, 56, 128, 60, 112, 104, 96, 64, 166, 78, 124, 96, 132, 72, 160, 96

Offset: 0

N. J. A. Sloane, Aug 09 2016

nonn

R. J. Mathar, Table of n, a(n) for n = 0..999
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016

Cf. A098151, A275633.

F:=(a,q,n)->mul(1-a*q^i,i=0..n-1); # This is (a;q)_n
M:=15;
# A098151:
THETA3:=(add((-1)^n*q^(3*n^2),n=-M..M)) /(add((-1)^n*q^(n^2),n=-M..M));
s1:=series(THETA3,q,80); seriestolist(%);
# A275632:
THETABAR3:=1+2*add( (F(q,q,n-1)*q^(n^2)) / (F(q^n,q,n)*(1-q^n)), n=1..M);
s2:=series(THETABAR3,q,80); seriestolist(%);
# A275633:
series((s1-s2)/8,q,80); seriestolist(%);

See Section 1 of Andrews (2016) or the Maple code below.

A304627 a(n) = [x^n] Product_{k>=1} (1 + x^k)(1 - x^(nk))/((1 - x^k)(1 + x^(nk))).

Original entry on oeis.org

1, 0, 2, 6, 12, 22, 38, 62, 98, 152, 230, 342, 502, 726, 1038, 1470, 2060, 2862, 3946, 5398, 7334, 9902, 13286, 17726, 23526, 31064, 40822, 53406, 69566, 90246, 116622, 150142, 192610, 246254, 313806, 398638, 504884, 637590, 802934, 1008446, 1263270, 1578526, 1967694, 2447062

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to partitions

Cf. A000065, A015128, A052839, A080054, A098151, A103258, A111133, A138526, A262968.

Table[SeriesCoefficient[Product[(1 + x^k) (1 - x^(n k))/((1 - x^k) (1 + x^(n k))) , {k, 1, n}], {x, 0, n}], {n, 0, 43}]
Table[SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n - 1}], {x, 0, n}], {n, 0, 43}]
Join[{1}, Table[SeriesCoefficient[EllipticTheta[4, 0, x^n]/EllipticTheta[4, 0, x], {x, 0, n}], {n, 43}]]
nmax = 43; CoefficientList[Series[1/EllipticTheta[4, 0, x] - 2 x/(1 - x), {x, 0, nmax}], x]

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

a(n) ~ exp(Pi*sqrt(n)) / (8*n). - Vaclav Kotesovec, May 19 2018

A335754 a(n) is the number of overpartitions of n where overlined parts are not divisible by 3 and non-overlined parts are congruent to 1 modulo 3.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 17, 23, 30, 39, 51, 66, 84, 107, 135, 168, 209, 259, 319, 391, 478, 581, 703, 849, 1022, 1226, 1466, 1748, 2078, 2465, 2917, 3443, 4055, 4765, 5588, 6540, 7640, 8908, 10368, 12047, 13973, 16182, 18712, 21604, 24906, 28673, 32964, 37846, 43397

Offset: 0

Jeremy Lovejoy, Jun 20 2020

nonn

			The 9 overpartitions counted by a(5) are: [5'], [4,1], [4,1'], [4',1], [4',1'], [2',1,1,1], [2',1',1,1], [1,1,1,1,1], [1',1,1,1,1].

J. Lovejoy, Asymmetric generalizations of Schur's theorem, in: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham.

Cf. A003105, A015128, A098151, A335755.

nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k-1)) * (1 + x^(3*k-2)) / (1 - x^(3*k-2)), {k, 1, nmax/3}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

G.f.: Product_{n>=1} (1+q^(3*n-1))*(1+q^(3*n-2))/(1-q^(3*n-2)).

a(n) ~ Gamma(1/3) * exp(2*Pi*sqrt(n)/3) / (2^(3/2) * sqrt(3) * Pi^(2/3) * n^(2/3)). - Vaclav Kotesovec, Jan 14 2021

A335755 a(n) is the number of overpartitions of n where overlined parts are not divisible by 3 and non-overlined parts are congruent to 2 modulo 3.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 9, 11, 14, 19, 24, 31, 39, 48, 61, 75, 93, 114, 139, 169, 205, 248, 298, 358, 428, 510, 607, 719, 851, 1005, 1182, 1389, 1628, 1904, 2225, 2592, 3015, 3501, 4058, 4698, 5429, 6264, 7216, 8302, 9538, 10944, 12541, 14351, 16403

Offset: 0

Jeremy Lovejoy, Jun 20 2020

nonn

			The 6 overpartitions counted by a(6) are: [5,1'], [5',1'], [4',2], [4',2'], [2,2,2], [2',2,2].

J. Lovejoy, Asymmetric generalizations of Schur's theorem, in: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham.

Cf. A003105, A015128, A098151, A335754.

nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k-1)) * (1 + x^(3*k-2)) / (1 - x^(3*k-1)), {k, 1, nmax/3}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

G.f.: Product_{n>=1} (1+q^(3*n-1))*(1+q^(3*n-2))/(1-q^(3*n-1)).

a(n) ~ Pi^(2/3) * exp(2*Pi*sqrt(n)/3) / (3*sqrt(2)*Gamma(1/3)*n^(5/6)). - Vaclav Kotesovec, Jan 14 2021

cp's OEIS Frontend

A261797 Expansion of Product_{k>=1} (1-x^(3*k))*(1-x^(5*k))/(1-x^k).

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A275633 Andrews's shadow difference function D_3(q).

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A100823 G.f.: Product_{k>0} (1+x^k)/((1-x^k)*(1+x^(3k))*(1+x^(5k))).

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A101230 Number of partitions of 2n in which both odd parts and parts that are multiples of 3 occur with even multiplicities. There is no restriction on the other even parts.

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A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.

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A252706 Expansion of phi(-q) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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A275632 Andrews's 4-shadow function THETA.BAR_3(q).

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A261797 Expansion of Product_{k>=1} (1-x^(3k))(1-x^(5*k))/(1-x^k).

A100823 G.f.: Product_{k>0} (1+x^k)/((1-x^k)(1+x^(3k))(1+x^(5k))).

A304627 a(n) = [x^n] Product_{k>=1} (1 + x^k)(1 - x^(nk))/((1 - x^k)(1 + x^(nk))).