A098884 -id:A098884 - cp's OEIS frontend

A109389 Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q.

1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, -1, 2, -2, 1, 0, 1, -2, 3, -3, 2, -1, 1, -3, 5, -5, 3, -1, 2, -5, 7, -7, 5, -3, 3, -7, 11, -11, 7, -4, 6, -11, 15, -15, 11, -7, 8, -15, 22, -22, 15, -10, 13, -22, 30, -30, 23, -16, 18, -30, 42, -42, 31, -22, 27, -43, 56, -56, 44, -33, 37, -57, 77, -77, 59, -45, 53, -79, 101, -101, 82, -64

Offset: 0

Michael Somos, Jun 26 2005

sign

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - Vaclav Kotesovec, Aug 31 2015

			q - q^13 - q^61 + q^73 - q^85 + q^97 - q^133 + 2*q^145 - 2*q^157 + q^169 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 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. 14.

Cf. A098884.

Cf. A081360 (m=2), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).

nmax = 100; CoefficientList[Series[Product[(1 + x^(3*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
QP = QPochhammer; s = QP[q]*(QP[q^6]/(QP[q^2]*QP[q^3])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *)

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

Euler transform of period 6 sequence [ -1, 0, 0, 0, -1, 0, ...].
G.f.: 1/(Product_{k>0} (1+x^(2k-1)+x^(4k-2))) = Product_{k>0} (1-x^(6k-1))(1-x^(6k-5)) = Product_{k>0} (1-x^k+x^(2k)) (where 1-x+x^2 is 6th cyclotomic polynomial).
Given g.f. A(x), then B(x)=x*A(x^12) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=(v^2+u^4)*(v^2+w^4)-2*v^4*(1-v*u^2*w^2).
Expansion of G(x^6) * H(x) - x * G(x) * H(x^6) where G(), H() are Rogers-Ramanujan functions.
a(n) = (-1)^n*A098884(n).
a(n) ~ (-1)^n * exp(sqrt(n)*Pi/3) / (2*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
a(n) = -(1/n)*Sum_{k=1..n} A186099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 26 2017

A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 5, 5, 1, 0, 0, 2, 7, 5, 1, 0, 0, 3, 8, 6, 1, 0, 0, 5, 10, 6, 1, 0, 1

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Convolution of this sequence and A280456 is A098884.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - minor asymptotic term, (Pi/144 - 9/(4*Pi))/sqrt(2) = -0.49100125...

Cf. A262928, A147599, A281243, A281245.

Cf. A261612, A169975, A280454, A280456, A280457.

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      [0$5, 1, 0$4, -1, 1][1+irem(d, 12)]*d, d=
       numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 18 2017

nmax = 200; CoefficientList[Series[Product[(1 + x^(6*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 6] == 5, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/2)*Pi/3) / (2^(25/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/144 - 9/(4*Pi)) / sqrt(2*n)).

G.f.: Sum_{k>=0} x^(k*(3*k + 2)) / Product_{j=1..k} (1 - x^(6*j)). - Ilya Gutkovskiy, Nov 24 2020

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

Original entry on oeis.org

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

A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 2, 0, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 4, 5, 2, 0, 0, 1, 5, 7, 3, 0, 0, 1, 5, 8, 5, 1, 0, 1, 6, 10, 6, 1, 0, 1, 6, 12, 9, 2, 0, 1, 7, 14, 11, 3, 0, 1, 7, 16, 15, 5, 0, 1, 8, 19, 18, 7, 1, 1, 8, 21, 23, 10, 1, 1, 9, 24

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 6.

Convolution of A281244 and A280456 is A098884. - Vaclav Kotesovec, Jan 18 2017

			a(32) = 3 because we have [31, 1], [25, 7] and [19, 13].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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.
Index entries for related partition-counting sequences

Cf. A000700, A016921, A109701, A169975, A261612, A280454, A280457.

Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 105; CoefficientList[Series[Product[(1 + x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 6] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(6*k+1)).

a(n) ~ exp(Pi*sqrt(n)/(3*sqrt(2)))/(2*2^(5/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/(144*sqrt(2)) - 9/(4*sqrt(2)*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 18 2017

A056970 Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 13, 13, 15, 16, 17, 20, 21, 23, 25, 27, 30, 33, 36, 38, 42, 45, 49, 54, 57, 62, 67, 72, 79, 85, 92, 98, 106, 114, 123, 133, 141, 152, 163, 175, 189, 202, 216, 231, 248, 265, 284, 304, 323

Offset: 0

Eric W. Weisstein

Also number of partitions of n into parts equal to 2,5, or 11 mod 12 (Gollnitz's theorem). Example: a(18)=4 because we have [14,2,2], [11,5,2], [5,5,2,2,2,2] and [2,2,2,2,2,2,2,2,2]. - Emeric Deutsch, Apr 18 2006

			a(18)=4 because we have [16,2], [14,4], [11,5,2] and [10,8].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
K. Alladi, Going beyond the partition theorem of Goellnitz
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 101.
G. E. Andrews, K. Alladi, and B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, Journal für die reine und angewandte Mathematik (1995), Volume: 460, page 165-188.
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. Vol. 225 (1967), 154-190.
Eric Weisstein's World of Mathematics, Göllnitz's Theorem.

Cf. A047261, A096981, A097451, A098884.

a056970 n = p a047261_list n where
   p _  0     = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Nov 16 2012

g:=product((1+x^(2+6*j))*(1+x^(4+6*j))*(1+x^(5+6*j)),j=0..30): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..67); # Emeric Deutsch, Apr 18 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(irem(d, 12) in [2, 5, 11], d, 0)
      , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Oct 27 2015

max = 70; g[x_] := Product[(1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)), {j, 0, Floor[max/6]}]; CoefficientList[ Series[g[x], {x, 0, max}], x](* Jean-François Alcover, Nov 16 2011, after Emeric Deutsch *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[If[MatchQ[Mod[d, 12], 2|5|11], d, 0], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Dec 23 2015, after Alois P. Heinz *)

{a(n)= if(n<0, 0, polcoeff( 1/prod(k=1, n, 1-(k%3==2)*(k%12!=8)*x^k, 1+x*O(x^n)), n))} /* Michael Somos, Jul 24 2007 */

From Emeric Deutsch, Apr 18 2006: (Start)
G.f.: Product_{j >= 0} (1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)).
G.f.: 1/Product_{j >= 0} (1-x^(2+12j))(1-x^(5+12j))(1-x^(11+12j)).
(End)
Euler transform of period 12 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, ...]. - Michael Somos, Jul 24 2007
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(25/12) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

A096981 Number of partitions of n into parts congruent to {0, 1, 3, 5} mod 6.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 7, 10, 12, 15, 21, 25, 30, 39, 46, 56, 72, 85, 101, 125, 147, 175, 215, 252, 296, 356, 415, 487, 582, 676, 786, 927, 1072, 1244, 1460, 1682, 1939, 2255, 2588, 2976, 3446, 3942, 4510, 5189, 5916, 6751, 7739, 8797, 9999, 11406, 12927, 14657

Offset: 0

Noureddine Chair, Aug 19 2004

nonn

Also, number of partitions of n in which the distinct parts are prime to 3 and the unrestricted parts are multiples of 3.

The inverted graded parafermionic partition function. This g.f. is a generalization of A003105, A006950 and A096938

			a(11) = 15 because we can write 11 = 10+1 = 8+2+1 = 7+4 = 5+4+2 (parts do not contain multiple of 3) = 9+2 = 8+3 = 7+3+1 = 6+5 = 6+4+1 = 6+3+2 = 5+3+3 = 5+3+2+1 = 4+3+3+1 = 3+3+3+2.
1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 10*x^9 + ...
q^-5 + q^19 + q^43 + 2*q^67 + 2*q^91 + 3*q^115 + 5*q^139 + 6*q^163 + 7*q^187 + ...

T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], 23-June-2013.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, arXiv:hep-th/9710002, 1997.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919.

Cf. A047273, A056970, A097451, A098884.

a096981 = p $ tail a047273_list where
   p _  0         = 1
   p ks'@(k:ks) m = if k > m then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 19 2013

series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)), k=1..150), x=0,100);

CoefficientList[ Series[ Product[ 1/(1 - x^k + x^(2k) - x^(3k) + x^(4k) - x^(5k)), {k, 55}], {x, 0, 53}], x] (* Robert G. Wilson v, Aug 21 2004 *)
nmax = 100; CoefficientList[Series[x^3*QPochhammer[-1/x^2, x^3] * QPochhammer[-1/x, x^3]/((1 + x)*(1 + x^2) * QPochhammer[x^3, x^3]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / (eta(x + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 08 2012 */

Expansion of q^(5/24) * eta(q^2) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jun 08 2012
Euler transform of period 6 sequence [1, 0, 1, 0, 1, 1, ...]. - Vladeta Jovovic, Aug 20 2004
G.f.: 1/product_{k>=1}(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)) = Product_{k>=1}(1+x^(3*k-1))(1+x^(3*k-2))/(1-x^(3*k)).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(6)*n). - Vaclav Kotesovec, Aug 31 2015

Better definition from Vladeta Jovovic, Aug 20 2004
More terms from Robert G. Wilson v, Aug 21 2004
Incorrect b-file replaced by Vaclav Kotesovec, Aug 31 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

A227398 Expansion of chi(x^3) / chi(x) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 1, -1, 1, -2, 2, -3, 3, -3, 4, -5, 6, -7, 8, -9, 10, -12, 14, -16, 18, -20, 23, -26, 30, -34, 38, -42, 47, -53, 60, -67, 74, -82, 91, -102, 114, -126, 139, -153, 169, -187, 207, -228, 250, -274, 301, -331, 364, -399, 436, -476, 520, -569, 622, -679

Offset: 0

Michael Somos, Sep 20 2013

sign

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

			G.f. = 1 - x + x^2 - x^3 + x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 3*x^8 - 3*x^9 + ...
G.f. = 1/q - q^11 + q^23 - q^35 + q^47 - 2*q^59 + 2*q^71 - 3*q^83 + 3*q^95 + ...

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. A003105, A098884.

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - (-x)^k + x^(2 k), {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k + x^(2 k), {k, 1, n, 2}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 3, n, 6}] / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / (Product[ 1 + x^k, {k, 1, n, 6}] Product[ 1 + x^k, {k, 5, n, 6}]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 + (-x)^k, {k, 1, n, 3}] Product[ 1 + (-x)^k, {k, 2, n, 3}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^6] / QPochhammer[ -x, x^2], {x, 0, n}];

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

{a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); m = sqrtint(3*n + 1); polcoeff( eta(x^6 + A) / sum(k= -((m-1)\3), (m+1)\3, x^(k * (3*k - 2)), A), n))};

Expansion of f(-x^6) / f(x, x^5) in powers of x where f(,) is Ramanujan's general theta function.
Expansion of q^(1/12) * eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [-1, 1, 0, 0, -1, 0, -1, 0, 0, 1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098884.
G.f.: 1 / Product_{k>0} (1 - (-x)^k + x^(2*k)).
G.f.: Product_{k>0} (1 - x^(2*k - 1) + x^(4*k - 2)).
G.f.: Product_{k>0} ((1 + x^(6*k - 3)) / (1 + x^(2*k - 1))).
G.f.: 1 / Product_{k>0} ((1 + x^(6*k - 1)) * (1 + x^(6*k - 5))).
G.f.: Product_{k>0} (1  + (-x)^(3*k - 1)) * (1 + (-x)^(3*k - 2)).
G.f.: (Sum_{k in Z} (-1)^k * x^(3*k * (3*k-1))) / (Sum_{k in Z} x^(k * (3*k - 2))).
a(n) = (-1)^n * A003105(n). Convolution inverse of A098884.

A284092 Number of partitions of n into distinct parts 8k+1 or 8k+7.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 3, 5, 5, 3, 1, 0, 0, 2, 5, 7, 7, 5, 2, 0, 1, 3, 7, 11, 11, 7, 3, 1, 1, 5, 11, 15, 15, 11, 5, 1, 2, 7, 15, 22, 22, 15, 7, 2, 3, 11, 22, 30, 30, 22, 11, 4, 5, 15, 30, 42, 42

Offset: 0

Seiichi Manyama, Mar 20 2017

nonn

Convolution of A284093 and A284095.

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Cf. A035453, A284093, A284095.

Cf. Product_{k>0} (1 + x^(m*k - 1)) * (1 + x^(m*k - m + 1)): A003105 (m=3), A000700 (m=4), A203776 (m=5), A098884 (m=6), A281459 (m=7), this sequence (m=8).

CoefficientList[Series[Product[(1 + x^(8*k - 1)) * (1 + x^(8*k - 7)) , {k, 1, 81}], {x, 0, 81}], x] (* Indranil Ghosh, Mar 20 2017 *)

Vec(prod(k=1, 81, (1 + x^(8*k - 1)) * (1 + x^(8*k - 7))) + O(x^82)) \\ Indranil Ghosh, Mar 20 2017

G.f.: Product_{k>0} (1 + x^(8*k - 1)) * (1 + x^(8*k - 7)).

a(n) ~ exp(sqrt(n/3)*Pi/2) / (4*3^(1/4)*n^(3/4)) * (1 + (11*Pi/(192*sqrt(3)) - 3*sqrt(3)/(4*Pi))/sqrt(n)). - Vaclav Kotesovec, Mar 20 2017

cp's OEIS Frontend

A109389 Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q.

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A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).

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

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A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

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A056970 Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6.

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A096981 Number of partitions of n into parts congruent to {0, 1, 3, 5} mod 6.

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