A097451 -id:A097451 - cp's OEIS frontend

A035294 Number of ways to partition 2n into distinct positive integers.

1, 1, 2, 4, 6, 10, 15, 22, 32, 46, 64, 89, 122, 165, 222, 296, 390, 512, 668, 864, 1113, 1426, 1816, 2304, 2910, 3658, 4582, 5718, 7108, 8808, 10880, 13394, 16444, 20132, 24576, 29927, 36352, 44046, 53250, 64234, 77312, 92864, 111322, 133184, 159046

Offset: 0

N. J. A. Sloane, Bill Gosper

Also, number of partitions of 2n into odd numbers. - Vladeta Jovovic, Aug 17 2004
This sequence was originally defined as the expansion of sum ( q^n / product( 1-q^k, k=1..2*n), n=0..inf ). The present definition is due to Reinhard Zumkeller. Michael Somos points out that the equivalence of the two definitions follows from Andrews, page 19.
Also, number of partitions of 2n with max descent 1 and last part 1. - Wouter Meeussen, Mar 31 2013

			a(4)=6 [8=7+1=6+2=5+3=5+2+1=4+3+1=2*4].
G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 22*x^7 + 46*x^9 + ...
G.f. = q + q^49 + 2*q^97 + 4*q^145 + 6*q^193 + 10*q^241 + 15*q^289 + ...

G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A058686, A262987, A282893.
Cf. A078408, A078406, A078407.
Cf. A079122, A079126, A079124, A079125, A067953.
Cf. A005408.

import Data.MemoCombinators (memo2, integral)
a035294 n = a035294_list !! n
a035294_list = f 1 where
   f x = (p' 1 (x - 1)) : f (x + 2)
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < k then 0 else p' k (m - k) + p' (k + 2) m
-- Reinhard Zumkeller, Nov 27 2015

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(2*n, 2*n-1):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 11 2015

Table[Count[IntegerPartitions[2 n], q_ /; Union[q] == Sort[q]], {n, 16}];
Table[Count[IntegerPartitions[2 n], q_ /; Count[q, _?EvenQ] == 0], {n, 16}];
Table[Count[IntegerPartitions[2 n], q_ /; Last[q] == 1 && Max[q - PadRight[Rest[q], Length[q]]] <= 1 ], {n, 16}];
(* Wouter Meeussen, Mar 31 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] /QPochhammer[ x], {x, 0, 2 n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 06 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^(8*k+1)) * (1+x^(8*k+2))^2 * (1+x^(8*k+3))^2 * (1+x^(8*k+4))^3 * (1+x^(8*k+5))^2 * (1+x^(8*k+6))^2 * (1+x^(8*k+7)) * (1+x^(8*k+8))^3, {k,0,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 06 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2n, 2n-1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

{a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))};/* Michael Somos, Nov 01 2005 */

a(n) = A000009(2*n). - Michael Somos, Mar 03 2003
Expansion of Sum_{n >= 0} q^n / Product_{k = 1..2*n} (1 - q^k).
a(n) = T(2*n, 0), T as defined in A026835.
G.f.: Product_{i >= 0} ((1 + x^(8*i + 1)) * (1 + x^(8*i + 2))^2 * (1 + x^(8*i + 3))^2 * (1 + x^(8*i + 4))^3 * (1 + x^(8*i + 5))^2 * (1 + x^(8*i + 6))^2 * (1 + x^(8*i + 7)) * (1 + x^(8*i + 8))^3). - Vladeta Jovovic, Oct 10 2004
G.f.: (Sum_{k>=0} x^A074378(k)) / (Product_{k>0} (1 - x^k)) = f( x^3, x^5) / f(-x, -x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 01 2005
Euler transform of period 16 sequence [1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 1, 1, 0, ...]. - Michael Somos, Dec 17 2002
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015
a(n) = A000041(n) + A282893(n). - Michael Somos, Feb 24 2017
Convolution with A000041 is A058696. - Michael Somos, Feb 24 2017
Convolution with A097451 is A262987. - Michael Somos, Feb 24 2017
G.f.: 1/(1 - x)*Sum_{n>=0} x^floor((3*n+1)/2)/Product_{k = 1..n} (1 - x^k). - Peter Bala, Feb 04 2021
G.f.: Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5))/(1 - q^n). - Peter Bala, Dec 30 2024

A047228 Numbers that are congruent to {2, 3, 4} mod 6.

Original entry on oeis.org

2, 3, 4, 8, 9, 10, 14, 15, 16, 20, 21, 22, 26, 27, 28, 32, 33, 34, 38, 39, 40, 44, 45, 46, 50, 51, 52, 56, 57, 58, 62, 63, 64, 68, 69, 70, 74, 75, 76, 80, 81, 82, 86, 87, 88, 92, 93, 94, 98, 99, 100, 104, 105, 106, 110, 111, 112, 116, 117, 118, 122, 123, 124

Offset: 1

N. J. A. Sloane

In other words, numbers that are divisible by 2 or by 3, but not by 6 (sorted). - David James Sycamore, Aug 22 2023

			From _David A. Corneth_, Aug 22 2023: (Start)
10 is in the sequence as 10 == 4 (mod 6) and 4 is in {2, 3, 4}.
11 is not in the sequence as 11 == 5 (mod 6) and 5 is not in {2, 3, 4}. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A007310, A047241, A047261, A047273, A097451.

a047228 n = a047228_list !! (n-1)
a047228_list = 2 : 3 : 4 : map (+ 6) a047228_list
-- Reinhard Zumkeller, Feb 19 2013

[n: n in [0..120] | n mod 6 in [2..4]]; // Vincenzo Librandi, Jan 05 2013

A047228:=n->2*n-1-cos(2*n*Pi/3)+sin(2*n*Pi/3)/sqrt(3): seq(A047228(n), n=1..100); # Wesley Ivan Hurt, Jun 13 2016

Select[Range[0, 150], MemberQ[{2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)

a(n) = 6*((n-1)\3) + 2 + (n-1)%3 \\ David A. Corneth, Aug 22 2023

nxt(n) = if(n%3 == 1, n+4, n+1) \\ David A. Corneth, Aug 22 2023

From Paul Barry, Sep 01 2009: (Start)
G.f.: (2+x+x^2+2*x^3)/(1-x-x^3+x^4).
a(n) = 2*n-1-cos(2*n*Pi/3)+sin(2*n*Pi/3)/sqrt(3). (End) [adapted for offset 1 by Wesley Ivan Hurt, Jun 13 2016]
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(3k) = 6k-2, a(3k-1) = 6k-3, a(3k-2) = 6k-4. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (4*sqrt(3)-3)*Pi/36. - Amiram Eldar, Dec 16 2021
E.g.f.: 2 + exp(x)*(2*x - 1) - exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Jul 26 2024

A098884 Number of partitions of n into distinct parts in which each part is congruent to 1 or 5 mod 6.

Original entry on oeis.org

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, 71

Offset: 0

Noureddine Chair, Oct 14 2004

nonn

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

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

			E.g. a(25)=5 because 25=19+5+1=17+7+1=13+7+5=13+11+1.
G.f. = 1 + x + x^5 + x^6 + x^7 + x^8 + x^11 + 2*x^12 + 2*x^13 + x^14 + x^16 + ...
G.f. = q + q^13 + q^61 + q^73 + q^85 + q^97 + q^133 + 2*q^145 + 2*q^157 + q^169 + ...

Reinhard Zumkeller and Vaclav Kotesovec, Table of n, a(n) for n = 0..20000 (terms 0..300 from Reinhard Zumkeller)
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A109389, A227398.

Cf. A007310, A056970, A096981, A097451.

a098884 = p a007310_list where
   p _  0     = 1
   p (k:ks) m = if k > m then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Feb 19 2013

series(product((1+x^(6*k-1))*(1+x^(6*k-5)),k=1..100),x=0,100);

a[ n_] := SeriesCoefficient[ Product[ 1 - (-x)^k + x^(2 k), {k, n}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k + x^(2 k), {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] / Product[ 1 + x^k, {k, 3, n, 6}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 6}] Product[ 1 + x^k, {k, 5, n, 6}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ -x^3, x^6], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 26 2005 */

{a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); m = sqrtint(3*n + 1); polcoeff( sum(k= -((m-1)\3), (m+1)\3, x^(k * (3*k - 2)), A) / eta(x^6 + A), n))}; /* Michael Somos, Sep 20 2013 */

Expansion of chi(x) / chi(x^3) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 20 2013
Expansion of f(x^1, x^5) / f(-x^6) in powers of x where f(,) is a Ramanujan theta function. - Michael Somos, Sep 20 2013
Expansion of G(x^6) * H(-x) + x * G(-x) * H(x^6) where G() (A003114), H() (A003106) are Rogers-Ramanujan functions.
Expansion of q^(-1/12) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)^2) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 0, 0, 1, 0, 1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Jun 26 2005
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 A227398. - Michael Somos, Sep 20 2013
G.f.: Product_{k>0} (1 - (-x)^k + x^(2*k)).
G.f.: 1 / Product_{k>0} (1 - x^(2*k - 1) + x^(4*k - 2)).
G.f.: 1 / Product_{k>0} ((1 + x^(6*k - 3)) / (1 + x^(2*k - 1))).
G.f.: Product_{k>0} ((1 + x^(6*k - 1)) * (1 + x^(6*k - 5))).
G.f.: 1 / Product_{k>0} (1  + (-x)^(3*k - 1)) * (1 + (-x)^(3*k - 2)).
G.f.: (Sum_{k in Z} x^(k * (3*k - 2))) / (Sum_{k in Z} (-1)^k * x^(3*k * (3*k-1))).
A109389(n) = (-1)^n * a(n). Convolution inverse of A227398.
a(n) ~ exp(sqrt(n)*Pi/3)/ (2*sqrt(6)*n^(3/4)) * (1 + (Pi/72 - 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 30 2015, extended Jan 18 2017

Typo in Maple program fixed by Vaclav Kotesovec, Nov 15 2016

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

A262987 Expansion of f(-x, -x^5) * f(x^3, x^5) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 3, 6, 11, 19, 33, 53, 86, 134, 205, 309, 460, 672, 974, 1394, 1975, 2773, 3863, 5333, 7316, 9964, 13484, 18140, 24269, 32288, 42751, 56331, 73888, 96503, 125529, 162635, 209939, 270027, 346123, 442213, 563205, 715110, 905361, 1142998, 1439098, 1807175

Offset: 0

Michael Somos, Oct 06 2015

nonn

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

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 33*x^6 + 53*x^7 + ...
G.f. = q^5 + q^21 + 3*q^37 + 6*q^53 + 11*q^69 + 19*q^85 + 33*q^101 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035294, A097451, A132217, A262160.

a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n}];
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x, -x^5]*f[x^3, x^5]/f[-x, -x^2]^2, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 31 2018 *)

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

Expansion of (psi(x^6) / psi(x) + psi(x^6) / psi(-x)) / 2 in powers of x^2 where psi() is a Ramanujan theta function.
Euler transform of period 48 sequence [1, 2, 3, 2, 2, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 0, 1, 1, 3, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 2, 2, 3, 2, 1, 0, ...].
a(n) = A132217(2*n) = A262160(2*n).
Convolution product of A035294 and A097451.
a(n) ~ exp(sqrt(n)*Pi)/(8*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2015

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.

A262146 Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 4, 8, 15, 25, 42, 68, 107, 166, 253, 377, 557, 811, 1166, 1661, 2344, 3275, 4543, 6253, 8544, 11600, 15653, 20994, 28011, 37178, 49100, 64550, 84489, 110115, 142951, 184867, 238196, 305844, 391391, 499244, 634865, 804925, 1017610, 1282957, 1613195

Offset: 0

Michael Somos, Oct 06 2015

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 + 8*x^3 + 15*x^4 + 25*x^5 + 42*x^6 + 68*x^7 + ...
G.f. = q^13 + 2*q^29 + 4*q^45 + 8*q^61 + 15*q^77 + 25*q^93 + 42*q^109 + ...

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. A078408, A097451, A132217, A262160.

a[ n_] := SeriesCoefficient[ - x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n + 1}];

{a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( - eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};

Expansion of - (psi(x^6) / psi(x) - psi(x^6) / psi(-x)) / (2 * x) in powers of x^2 where psi() is a Ramanujan theta function.
Euler transform of period 48 sequence [ 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 3, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 2, 2, 1, 0, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 2, 2, 1, 2, 0, ...].
a(n) = A132217(2*n + 1) = - A262160(2*n + 1).
Convolution product of A097451 and A078408.
a(n) ~ exp(Pi*sqrt(n)) / (2^(7/2) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Mar 31 2018

cp's OEIS Frontend

A035294 Number of ways to partition 2n into distinct positive integers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A047228 Numbers that are congruent to {2, 3, 4} mod 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

A098884 Number of partitions of n into distinct parts in which each part is congruent to 1 or 5 mod 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A262987 Expansion of f(-x, -x^5) * f(x^3, x^5) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Views