A070047 -id:A070047 - cp's OEIS frontend

A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6.

1, 0, 1, 1, 2, 1, 3, 2, 5, 4, 7, 6, 11, 9, 15, 14, 22, 20, 31, 29, 43, 41, 58, 57, 80, 78, 106, 107, 142, 143, 188, 191, 247, 253, 321, 332, 418, 432, 537, 561, 690, 721, 880, 924, 1118, 1178, 1412, 1493, 1781, 1884, 2231, 2370, 2789, 2965, 3472, 3698, 4309, 4596

Offset: 0

Vladeta Jovovic, Aug 23 2004

Number of partitions of n in which no part is 1, no part appears more than twice and no two parts differ by 1. Example: a(6)=3 because we have [6],[4,2] and [3,3]. - Emeric Deutsch, Feb 16 2006

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

			a(8)=5 because we have [8],[44],[422],[332] and [2222].
G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...
G.f. = q^7 + q^55 + q^79 + 2*q^103 + q^127 + 3*q^151 + 2*q^175 + 5*q^199 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, Exercise 7.9.

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A070047.

Cf. A047228, A056970, A096981, A098884.

a097451 n = p a047228_list n where
   p _  0         = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 16 2012

g:=1/product((1-x^(2+6*j))*(1-x^(3+6*j))*(1-x^(4+6*j)),j=0..15): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=0..67); # Emeric Deutsch, Feb 16 2006

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - Boole[ OddQ[ Quotient[ k + 1, 3]]] x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Sep 24 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^3] QPochhammer[ x^6] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Sep 24 2013 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ( (k+1)\3 % 2) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 24 2013 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 24 2013 */

Euler transform of period 6 sequence [ 0, 1, 1, 1, 0, 0, ...].
G.f.: 1/Product_{j>=0} ((1-x^(2+6j))(1-x^(3+6j))(1-x^(4+6j))). - Emeric Deutsch, Feb 16 2006
Expansion of psi(x^3) / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 24 2013
Expansion of q^(-7/24) * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Sep 24 2013
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
Expansion of f(-x, -x^5) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015

More terms from Emeric Deutsch, Feb 16 2006

A108961 Number of partitions that are "2-close" to being self-conjugate.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 7, 9, 12, 16, 20, 26, 33, 41, 51, 64, 79, 97, 119, 144, 175, 212, 254, 305, 365, 434, 516, 612, 722, 851, 1002, 1174, 1375, 1607, 1872, 2179, 2531, 2933, 3395, 3923, 4524, 5211, 5994, 6881, 7891, 9038, 10334, 11804, 13467, 15341, 17460, 19849

Offset: 0

John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005

nonn

Let (a1,a2,a3,...ad:b1,b2,b3,...bd) be the Frobenius symbol for a partition pi of n. Then pi is m-close to being self-conjugate if for all k, |ak-bk| <= m.

Convolution of A070048 and A035457. - Vaclav Kotesovec, Nov 13 2016

			1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 12*x^8 + 16*x^9 + 20*x^10 + ...
1/q + q^23 + 2*q^47 + 3*q^71 + 3*q^95 + 5*q^119 + 7*q^143 + 9*q^167 + 12*q^191 + ...

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
D. M. Bressoud, Extension of the partition sieve, J. Number Theory 12 no. 1 (1980), 87-100.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Wikipedia, Bailey pair

Cf. A000700 for m=0 (self-conjugate), A070047 for m=1, A108962 for m=3, A271661 for m=4, A280937 for m=5, A280938 for m=6.

nmax = 50; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(2*k)) / (1 + x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x + A) * eta(x^8 + A)), n))} /* Michael Somos, Oct 17 2006 */

Define the Dedekind eta function = q^1/24. Product(1-q^k), k >=1. Then the number of m-close partitions is q^(1/24).(m+2)^2/(1.(2m+4)) (where m denotes eta(q^m)).
Expansion of q^(1/24) * eta(q^4)^2 / (eta(q) * eta(q^8)) in powers of q. - Michael Somos, Oct 17 2006
Expansion of chi(x^2) * chi(x) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function (see A000700). - Michael Somos, Oct 17 2006 [corrected by Peter Bala, Oct 09 2023]
Euler transform of period 8 sequence [ 1, 1, 1, -1, 1, 1, 1, 0, ...]. - Michael Somos, Oct 17 2006
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k)) / (1 + x^(4*k)). - Michael Somos, Oct 17 2006
a(n) ~ Pi * BesselI(1, Pi * sqrt(5*(24*n-1)/2)/12) / (2*sqrt((24*n-1)/5)) ~ 5^(1/4) * exp(sqrt(5*n/3)*Pi/2) / (2^(5/2) * 3^(1/4) * n^(3/4)) * (1 -(3*sqrt(3) / (4*Pi*sqrt(5)) + Pi*sqrt(5)/(96*sqrt(3)))/sqrt(n) + (5*Pi^2/55296 - 9/(32*Pi^2) + 5/128)/n). - Vaclav Kotesovec, Nov 13 2016, extended Jan 11 2017

A108962 Number of partitions that are "3-close" to being self-conjugate.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 9, 11, 16, 20, 28, 34, 47, 57, 75, 92, 119, 143, 183, 220, 277, 332, 412, 491, 605, 718, 874, 1036, 1252, 1475, 1772, 2082, 2483, 2909, 3450, 4027, 4755, 5533, 6499, 7545, 8826, 10213, 11904, 13741, 15955, 18372, 21262, 24422

Offset: 0

John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005

nonn

Let (a1,a2,a3,...ad:b1,b2,b3,...bd) be the Frobenius symbol for a partition pi of n. Then pi is m-close to being self-conjugate if for all k, |ak-bk| <= m.

			1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 11*x^7 + 16*x^8 + 20*x^9 + 28*x^10 + ...
1/q + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 5*q^119 + 9*q^143 + 11*q^167 + 16*q^191 + ...

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
D. M. Bressoud, Extension of the partition sieve, J. Number Theory 12 no. 1 (1980), 87-100.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Wikipedia, Bailey pair

Cf. A000700 for m=0 (self-conjugate), A070047 for m=1, A108961 for m=2, A271661 for m=4, A280937 for m=5, A280938 for m=6.

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^2 / ((1 - x^k) * (1 - x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A)^2 / (eta(x + A) * eta(x^10 + A)), n))} /* Michael Somos, Jun 08 2012 */

Define the Dedekind eta function = q^1/24. Product(1-q^k), k >=1. Then the number of m-close partitions is q^(1/24).(m+2)^2/(1.(2m+4)) (where m denotes eta(q^m)).
From Vaclav Kotesovec, Nov 13 2016, extended Jan 11 2017: (Start)
a(n, m) ~ exp(Pi*sqrt((2*m+1)*n/(3*(m+2)))) * (2*m+1)^(1/4) / (2*3^(1/4)*(m+2)^(3/4)*n^(3/4)).
For m=3, a(n) ~ 7^(1/4) * exp(sqrt(7*n/15)*Pi) / (2*3^(1/4)*5^(3/4)*n^(3/4)) * (1 -(3*sqrt(15)/(8*Pi*sqrt(7)) + Pi*sqrt(7)/(48*sqrt(15)))/sqrt(n) + (7*Pi^2/69120 - 225/(896*Pi^2) + 5/128)/n).
(End)
a(n) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)*7/10)) / (5*sqrt((24*n-1)/7)). - Vaclav Kotesovec, Jan 11 2017

A271661 Expansion of phi(-x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 13, 18, 24, 32, 42, 55, 71, 91, 116, 147, 185, 231, 288, 357, 440, 540, 661, 807, 980, 1186, 1432, 1724, 2069, 2476, 2956, 3521, 4183, 4958, 5865, 6923, 8155, 9587, 11251, 13180, 15411, 17990, 20967, 24399, 28348, 32886, 38098, 44075

Offset: 0

Michael Somos, Apr 11 2016

nonn

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

Partitions which are "4-close" to being self-conjugate; see A108961 for the definition. - Arvind Ayyer, Apr 13 2021

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 13*x^7 + 18*x^8 + ...
G.f. = q^-1 + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 9*q^143 + 13*q^167 + ..

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988. See page 6 equation 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
D. M. Bressoud, Extension of the partition sieve, J. Number Theory 12 no. 1 (1980), 87-100.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Wikipedia, Bailey pair

Cf. A000700, A070047, A108961, A108962, A280937, A280938.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^6] / QPochhammer[ x], {x, 0, n}];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x, x, 2 k] // FunctionExpand, {k, 0, Sqrt@n}], {x, 0, n}]];
nmax = 50; CoefficientList[Series[Product[(1-x^(6*k)) / ((1-x^k) * (1+x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 18 2016 *)

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

Expansion of f(x^2, x^4) / psi(-x) in powers of x where psi(), f(, ) are Ramanujan theta functions.
Expansion of q^(1/24) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 0, ...].
G.f.: Sum_{k>=0} x^(k^2) (-x, x^2)k / (x){2*k}.
a(n) ~ Pi * BesselI(1, Pi*sqrt(24*n-1)/(4*sqrt(3))) / sqrt(24*n-1) ~ exp(sqrt(n/2)*Pi) / (2^(7/4)*sqrt(3)*n^(3/4)) * (1 - (3/(4*Pi) + Pi/48)/sqrt(2*n) + (5/128 - 15/(64*Pi^2) + Pi^2/9216)/n). - Vaclav Kotesovec, Apr 18 2016, extended Jan 10 2017

A280937 Expansion of Product_{k>=1} ((1 - x^(7(2k-1))) * (1 - x^(7*k)) / (1 - x^k)).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 13, 20, 26, 36, 46, 63, 79, 105, 132, 171, 213, 273, 336, 425, 522, 650, 793, 981, 1188, 1456, 1756, 2136, 2563, 3098, 3698, 4443, 5285, 6312, 7477, 8891, 10489, 12415, 14599, 17206, 20165, 23678, 27659, 32363, 37698, 43958, 51058, 59361

Offset: 0

Vaclav Kotesovec, Jan 11 2017

nonn

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Wikipedia, Bailey pair.

Cf. A000700, A070047, A108961, A108962, A271661, A280938.

nmax = 50; CoefficientList[Series[Product[(1-x^(7*(2*k-1))) * (1-x^(7*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2*Pi * BesselI(1, Pi/6 * sqrt(11*(24*n-1)/14)) / (7*sqrt((24*n-1)/11)).

a(n) ~ exp(Pi * sqrt(11*n/21)) * 11^(1/4) / (2 * 3^(1/4) * 7^(3/4) * n^(3/4)) * (1 -(3*sqrt(21)/(8*Pi*sqrt(11)) + Pi*sqrt(11)/(48*sqrt(21)))/sqrt(n) + (11*Pi^2/96768 - 315/(1408*Pi^2) + 5/128)/n).

A280938 Expansion of Product_{k>=1} (1 - x^(8(2k-1))) * (1 - x^(8*k)) / (1 - x^k).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 20, 28, 38, 50, 67, 87, 113, 146, 187, 237, 301, 378, 473, 590, 732, 903, 1113, 1364, 1666, 2030, 2464, 2981, 3600, 4332, 5201, 6229, 7442, 8869, 10551, 12521, 14829, 17531, 20684, 24357, 28638, 33607, 39375, 46062, 53798, 62736

Offset: 0

Vaclav Kotesovec, Jan 11 2017

nonn

In general, if r>=2 and g.f. = Product_{k>=1} (1-x^(r*(2*k-1))) * (1-x^(r*k)) / (1-x^k), then
a(n, r) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)*(2*r-3)/(2*r))) / (r*sqrt((24*n-1)/(2*r-3))).
a(n, r) ~ exp(Pi * sqrt((2/3 - 1/r)*n)) * (2*r-3)^(1/4) / (2 * 3^(1/4) * r^(3/4) * n^(3/4)) * (1 -(3*sqrt(3*r)/(8*Pi*sqrt(2*r-3)) + Pi*sqrt(2*r-3)/(48*sqrt(3*r))) / sqrt(n) + (Pi^2*(2*r-3)/(13824*r) - 45*r/(128*Pi^2*(2*r-3)) + 5/128)/n).

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Wikipedia, Bailey pair.

Cf. A000700 (r=2), A070047 (r=3), A108961 (r=4), A108962 (r=5), A271661 (r=6), A280937 (r=7).

nmax = 50; CoefficientList[Series[Product[(1-x^(8*(2*k-1))) * (1-x^(8*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ Pi * BesselI(1, Pi * sqrt(13*(24*n-1))/24) / (4*sqrt((24*n-1)/13)).

a(n) ~ exp(Pi*sqrt(13*n/6)/2) * 13^(1/4) / (2^(13/4) * 3^(1/4) * n^(3/4)) * (1 -(3*sqrt(3)/(2*Pi*sqrt(26)) + Pi*sqrt(13)/(96*sqrt(6)))/sqrt(n) + (13*Pi^2/110592 - 45/(208*Pi^2) + 5/128)/n).

A233006 Expansion of psi(x) / f(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 3, 2, 0, 3, 1, 0, 5, 3, 0, 5, 2, 0, 8, 5, 0, 8, 4, 0, 12, 7, 0, 12, 6, 0, 19, 11, 0, 19, 9, 0, 27, 15, 0, 28, 14, 0, 39, 22, 0, 41, 20, 0, 55, 31, 0, 58, 29, 0, 77, 43, 0, 82, 41, 0, 106, 58, 0, 113, 57, 0, 145, 80, 0, 156

Offset: 0

Michael Somos, Dec 03 2013

nonn

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

			G.f. = 1 + x + x^3 + 2*x^6 + x^7 + x^9 + x^10 + 3*x^12 + 2*x^13 + 3*x^15 + ...
G.f. = q + q^9 + q^25 + 2*q^49 + q^57 + q^73 + q^81 + 3*q^97 + 2*q^105 + ...

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. A070047, A097451.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8) QPochhammer[ x^6]), {x, 0, n}];

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

Expansion of q^(-1/8) * eta(q^2)^2 / (eta(q) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 1, -1, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = (3/2)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A070047.
G.f.: Product_{k>0} (1 + x^k) / (1 + x^(2*k) + x^(4*k)).
a(3*n) = A070047(n). a(3*n + 1) = A097451(n). a(3*n + 2) = 0.

A294199 Number of partitions of n into powers of 2 such that 1 and 2 cannot both be parts of a particular partition, and 4 and 8 cannot both be parts of a particular partition, and 16 and 32, and so on.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 2, 6, 4, 8, 4, 9, 5, 10, 5, 13, 8, 16, 8, 18, 10, 20, 10, 24, 14, 28, 14, 30, 16, 32, 16, 38, 22, 44, 22, 48, 26, 52, 26, 60, 34, 68, 34, 72, 38, 76, 38, 85, 47, 94, 47, 99, 52, 104, 52, 114, 62, 124, 62, 129, 67, 134, 67, 147, 80, 160

Offset: 0

Vaclav Kotesovec, Oct 24 2017

nonn

			a(10) = 8 where the partitions are the following: 8+2, 8+1+1, 4+4+2, 4+2+2+2, 4+4+1+1, 4+1+1+1+1+1+1, 2+2+2+2+2, 1+1+1+1+1+1+1+1+1+1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Bin Lan and James A. Sellers, Properties of a Restricted Binary Partition Function a la Andrews and Lewis, #A23 INTEGERS 15 (2015), p.2.

Cf. A070047.

nmax = 20; CoefficientList[Series[Product[(1-x^(3*2^(2*k-2)))/(1-x^(2^(k-1))), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[1] = 1; a[2] = 2; a[3] = 1; Flatten[{1, 1, 2, 1, Table[If[EvenQ[n], a[n] = a[n-2] + a[Floor[n/4]], a[n] = a[n-1] - a[n-2]], {n, 4, 100}]}]

G.f.: Product_{k>=1} (1 - x^(2^(2*k-2) + 2^(2*k-1))) / ((1 - x^(2^(2*k-2))) * (1 - x^(2^(2*k-1)))).
G.f.: Product_{k>=1} (1 - x^(3*2^(2*k-2))) / (1 - x^(2^(k-1))).
For n>=1 a(2*n) = a(2*n-2) + a([n/2]).
For n>=1 a(2*n+1) = a(2*n) - a(2*n-1).

cp's OEIS Frontend

A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6.

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A108961 Number of partitions that are "2-close" to being self-conjugate.

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A108962 Number of partitions that are "3-close" to being self-conjugate.

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A271661 Expansion of phi(-x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

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A280937 Expansion of Product_{k>=1} ((1 - x^(7*(2*k-1))) * (1 - x^(7*k)) / (1 - x^k)).

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A280938 Expansion of Product_{k>=1} (1 - x^(8*(2*k-1))) * (1 - x^(8*k)) / (1 - x^k).

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A233006 Expansion of psi(x) / f(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.

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A280937 Expansion of Product_{k>=1} ((1 - x^(7(2k-1))) * (1 - x^(7*k)) / (1 - x^k)).

A280938 Expansion of Product_{k>=1} (1 - x^(8(2k-1))) * (1 - x^(8*k)) / (1 - x^k).