A000025 -id:A000025 - cp's OEIS frontend

A053251 Coefficients of the '3rd-order' mock theta function psi(q).

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 19, 22, 24, 27, 31, 34, 37, 42, 46, 51, 57, 62, 68, 76, 83, 91, 101, 109, 120, 132, 143, 156, 171, 186, 202, 221, 239, 259, 283, 306, 331, 360, 388, 420, 455, 490, 529, 572, 616, 663, 716, 769, 827

Offset: 0

Dean Hickerson, Dec 19 1999

Number of partitions of n into odd parts such that if a number occurs as a part then so do all smaller positive odd numbers.
Number of ways to express n as a partial sum of 1 + [1,3] + [1,5] + [1,7] + [1,9] + .... E.g., a(6)=2 because we have 6 = 1+1+1+1+1+1 = 1+3+1+1. - Jon Perry, Jan 01 2004
Also number of partitions of n such that the largest part occurs exactly once and all the other parts occur exactly twice. Example: a(9)=4 because we have [9], [7,1,1], [5,2,2] and [3,2,2,1,1]. - Emeric Deutsch, Mar 08 2006
Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 < d2/2 < ... < dm/m. - Seiichi Manyama, Mar 17 2018
For Emeric Deutsch's comment above, (1) this appears to be an alternately equal case of A122130, (2) the ordered version (compositions) is A239327, (3) allowing any length gives A351006, (4) the even-length version is A351007. - Gus Wiseman, Feb 25 2022

			q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ...
From _Seiichi Manyama_, Mar 17 2018: (Start)
n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m)
--+--------------------------+-------------------------
1 | (1)                      | (1)
2 | (2)                      | (2)
3 | (3)                      | (3)
4 | (4)                      | (4)
  | (1, 3)                   | (1, 3/2)
5 | (5)                      | (5)
  | (1, 4)                   | (1, 2)
6 | (6)                      | (6)
  | (1, 5)                   | (1, 5/2)
7 | (7)                      | (7)
  | (1, 6)                   | (1, 3)
  | (2, 5)                   | (2, 5/2)
8 | (8)                      | (8)
  | (1, 7)                   | (1, 7/2)
  | (2, 6)                   | (2, 3)
9 | (9)                      | (9)
  | (1, 8)                   | (1, 4)
  | (2, 7)                   | (2, 7/2)
  | (1, 3, 5)                | (1, 3/2, 5/3) (End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.13).
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053250, A053252, A053253, A053254, A053255.
Cf. A003475.
Cf. A035363, A035457, A122129, A122130, A122134, A122135, A351003, A351005.

f:=n->q^(n^2)/mul((1-q^(2*i+1)),i=0..n-1); add(f(i),i=1..6);
# second Maple program:
b:= proc(n, i) option remember; (s-> `if`(n>s, 0, `if`(n=s, 1,
      b(n, i-1)+b(n-i, min(n-i, i-1)))))(i*(i+1)/2)
    end:
a:= n-> `if`(n=0, 0, add(b(j, min(j, n-2*j-1)), j=0..iquo(n, 2))):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018

Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 100}]
(* Second program: *)
b[n_, i_] := b[n, i] = Function[s, If[n > s, 0, If[n == s, 1, b[n, i - 1] + b[n - i, Min[n - i, i - 1]]]]][i*(i + 1)/2];
a[n_] := If[n==0, 0, Sum[b[j, Min[j, n-2*j-1]], {j, 0, Quotient[n, 2]}]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 17 2018, after Alois P. Heinz *)

{ n=20; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]+2*i-1)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry

{a(n) = local(t); if(n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k-1) / (1 - x^(2*k-1)) + O(x^(n-(k-1)^2+1))), n))} /* Michael Somos, Sep 04 2007 */

G.f.: psi(q) = Sum_{n>=1} q^(n^2) / ( (1-q)*(1-q^3)*...*(1-q^(2*n-1)) ).
G.f.: Sum_{k>=1} q^k*Product_{j=1..k-1} (1+q^(2*j)) (see the Fine reference, p. 58, Eq. (26,53)). - Emeric Deutsch, Mar 08 2006
a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 09 2019

More terms from Emeric Deutsch, Mar 08 2006

A040051 Parity of partition function A000041.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1

Offset: 0

N. J. A. Sloane

From M. V. Subbarao (m.v.subbarao(AT)ualberta.ca), Sep 05 2003: (Start)
Essentially this same question was raised by Ramanujan in a letter to P. A. MacMahon around 1920 (see page 1087, MacMahon's Collected Papers). With the help of Jacobi's triple product identity, MacMahon showed that p(1000) is odd (as he says, with five minutes work - there were no computers those days).
Now we know that among the first ten million values of p(n) 5002137 of them are odd. It is conjectured (T. R. Parkin and D. Shanks) that p(n) is equally often even and odd. Lower bound estimates for the number of times p(n) is even among the first N values of p(n) for any given N are known (Scott Ahlgren; and Nicolas, Rusza and Sárközy among others).
Earlier this year a remarkable result was proved by Boylan and Ahlgren (AMS ABSTRACT # 987-11-82) which says that beyond the three eighty-year old Ramanujan congruences - namely, p(5n+4), p(7n+5) and p(11n +6) being divisible respectively by 5,7 and 11 - there are no other simple congruences of this kind.
My 1966 conjecture that in every arithmetic progression r (mod s) for arbitrary integral r and s, there are infinitely many integers n for which p(n) is odd - with a similar statement for p(n) even - was proved for the even case by Ken Ono (1996) and for the odd case for all s up to 10^5 and for all s which are powers of 2 by Bolyan and Ono, 2002.
(End)
a(n) is also the parity of the trace Tr(n) = A183011(n), the numerator of the Bruinier-Ono formula for the partition function, if n >= 1. - Omar E. Pol, Mar 14 2012
Consider the diagram of the regions of n (see A206437). Then, in each odd-indexed region of n, fill each part of size k with k 1's. Then, in each even-indexed region of n, fill each part of size k with k 0's. The successive digits of row 1 of the diagram give the first n elements of this sequence, if n >= 1. - Omar E. Pol, May 02 2012

H. Gupta, A note on the parity of p(n), J. Indian Math. Soc. (N.S.) 10, (1946). 32-33. MR0020588 (8,566g)
K. M. Majumdar, On the parity of the partition function p(n), J. Indian Math. Soc. (N.S.) 13, (1949). 23-24. MR0030553 (11,13d)
M. V. Subbarao, A note on the parity of p(n), Indian J. Math. 14 (1972), 147-148. MR0357355 (50 #9823)

T. D. Noe, Table of n, a(n) for n = 0..1000
R. Blecksmith; J. Brillhart; I. Gerst, Parity results for certain partition functions and identities similar to theta function identities, Math. Comp. 48 (1987), no. 177, 29-38. MR0866096 (87k:11113).
Nicholas Eriksson, q-series, elliptic curves and odd values of the partition function, Int. J. Math. Math. Sci. 22 (1999), 55-65; MR 2001a:11175.
M. D. Hirschhorn, On the residue mod 2 and mod 4 of p(n), Acta Arith. 38 (1980/81), no. 2, 105-109. MR0604226 (82d:10025)
M. D. Hirschhorn, On the parity of p(n), II, J. Combin. Theory Ser. A 62 (1993), no. 1, 128-138.
M. D. Hirschhorn and M. V. Subbarao, On the parity of p(n), Acta Arith. 50 (1988), no. 4, 355-356.
O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).
P. A. MacMahon, The parity of p(n), the number of partitions of n, when n <= 1000, J. London Math. Soc., 1 (1926), 225-226.
Mircea Merca, New recurrences for Euler's partition function, Turkish J. Math. 41:5 (2017), pp. 1184-1190.
M. Newman, Periodicity modulo m and divisibility properties of the partition function, Trans. Amer. Math. Soc. 97 (1960), 225-236. MR0115981 (22 #6778)
M. Newman, Congruences for the partition function to composite moduli, Illinois J. Math. 6 1962 59-63. MR0140472 (25 #3892)
K. Ono, Parity of the partition function, Electron. Res. Announc. AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108.
Ivars Peterson, Ken Ono's and Nicholas Eriksson's work

Cf. A000041, A071640, A086144, A190938.

import Data.Bits (xor)
a040051 n = p 1 n :: Int where
   p _ 0 = 1
   p k m | k <= m = p k (m - k) `xor` p (k+1) m | k > m  = 0
-- Reinhard Zumkeller, Nov 15 2011

Table[ Mod[ PartitionsP@ n, 2], {n, 105}] (* Robert G. Wilson v, Mar 25 2011 *)

PARI
```
a(n)=if(n<0, 0, numbpart(n)%2)
```

a(n)=if(n<0, 0, polcoeff(1/eta(x+x*O(x^n)), n)%2)

a(n)=if(n<10^9, return(numbpart(n)%2)); my(r=n%4, u=select(k->k^2%32==8*r+1,[1..31]), st=u[1], m=n\4, s); u=[u[2]-u[1],u[3]-u[2],u[4]-u[3],u[1]+32-u[4]]; forstep(t=[1,3,7,5][r+1], sqrtint(32*m-1), u, k=t^2>>5; if(a(m-k), s++)); s%2 \\ Merca's algorithm, switching to direct computation for n less than 10^9. Very time-consuming but low memory use. - Charles R Greathouse IV, Jan 24 2018

from sympy import npartitions
def a(n): return npartitions(n)%2 # Indranil Ghosh, May 25 2017

a(n) = pp(n, 1), with Boolean pp(n, k) = if kReinhard Zumkeller, Sep 04 2003
a(n) = Pm(n,1) with Pm(n,k) = if kReinhard Zumkeller, Jun 09 2009
a(n) = A000035(A000041(n)). - Omar E. Pol, Aug 05 2013
a(n) = A000035(A000025(n)). - John M. Campbell, Jun 29 2016

A063995 Irregular triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2

Offset: 1

N. J. A. Sloane, Sep 19 2001

The rank of a partition is the largest part minus the number of parts.

The rows are symmetric: for every partition of rank r there is its conjugate with rank -r. [Joerg Arndt, Oct 07 2012]

			The partition 5 = 4+1 has largest summand 4 and 2 summands, hence has rank 4-2 = 2.
Triangle begins:
[ 1]                               1,
[ 2]                            1, 0, 1,
[ 3]                         1, 0, 1, 0, 1,
[ 4]                      1, 0, 1, 1, 1, 0, 1,
[ 5]                   1, 0, 1, 1, 1, 1, 1, 0, 1,
[ 6]                1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1,
[ 7]             1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1,
[ 8]          1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1,
[ 9]       1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1,
[10]    1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1,
[11] 1, 0, 1, 1, 2, ...
Row 20 is:
T(20, k) = 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 22, 30, 33, 40, 42, 48, 45, 48, 42, 40, 33, 30, 22, 20, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1; -19 <= k <= 19.
Another view of the table of p(n,m) = number of partitions of n with rank m, taken from Dyson (1969):
n\m -6 -5  -4  -3  -2  -1   0   1   2   3   4   5   6
-----------------------------------------------------
0   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,
1   0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  0,
2   0,  0,  0,  0,  0,  1,  0,  1,  0,  0,  0,  0,  0,
3   0,  0,  0,  0,  1,  0,  1,  0,  1,  0,  0,  0,  0,
4   0,  0,  0,  1,  0,  1,  1,  1,  0,  1,  0,  0,  0,
5   0,  0,  1,  0,  1,  1,  1,  1,  1,  0,  1,  0,  0,
6   0,  1,  0,  1,  1,  2,  1,  2,  1,  1,  0,  1,  0,
7   1,  0,  1,  1,  2,  1,  3,  1,  2,  1,  1,  0,  1,
...
The central triangle is the present sequence, the right-hand triangle is A105806. - _N. J. A. Sloane_, Jan 23 2020

Alois P. Heinz, Rows n = 1..145, flattened (first 72 rows from Reinhard Zumkeller)
G. E. Andrews, The number of smallest parts in the partitions of n. [Also Selected Works, p. 603, see N(m,n).] - _N. J. A. Sloane_, Dec 16 2013
A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4, (1954). 84-106. Math. Rev. 15,685d.
Alexander Berkovich and Frank G. Garvan, Some observations on Dyson's new symmetries of partitions, Journal of Combinatorial Theory, Series A 100.1 (2002): 61-93.
Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61. See Table 1.
Freeman J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169-180.

For the number of partitions of n with rank 0 (balanced partitions) see A047993.
Cf. A105806 (right half of triangle), A005408 (row lengths), A000041 (row sums), A047993 (central terms).
Cf. A000025.

import Data.List (sort, group)
a063995 n k = a063995_tabf !! (n-1) !! (n-1+k)
a063995_row n = a063995_tabf !! (n-1)
a063995_tabf = [[1], [1, 0, 1]] ++ (map
   (\rs -> [1, 0] ++ (init $ tail $ rs) ++ [0, 1]) $ drop 2 $ map
   (map length . group . sort . map rank) $ tail pss) where
      rank ps = maximum ps - length ps
      pss = [] : map (\u -> [u] : [v : ps | v <- [1..u],
                             ps <- pss !! (u - v), v <= head ps]) [1..]
-- Reinhard Zumkeller, Jul 24 2013

Table[ Count[ (First[ # ]-Length[ # ]& /@ IntegerPartitions[ k ]), # ]& /@ Range[ -k+1, k-1 ], {k, 16} ]

Sum_{k=-(n-1)..n-1} (-1)^k * T(n,k) = A000025(n). - Alois P. Heinz, Dec 20 2024

More terms from Vladeta Jovovic and Wouter Meeussen, Sep 19 2001

A101707 Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 2, 7, 6, 13, 11, 22, 22, 38, 39, 63, 69, 103, 114, 165, 189, 262, 301, 407, 475, 626, 733, 950, 1119, 1427, 1681, 2118, 2503, 3116, 3678, 4539, 5360, 6559, 7735, 9400, 11076, 13372, 15728, 18886, 22184, 26501, 31067, 36947, 43242, 51210, 59818, 70576, 82291, 96750

Offset: 0

Emeric Deutsch, Dec 12 2004

nonn

a(n) + A101708(n) = A064173(n).

			a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).
From _Gus Wiseman_, Feb 07 2021: (Start)
Also the number of integer partitions of n into an even number of parts, the greatest of which is odd. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot) are:
  11   .  31     32   33       52     53         54       55
          1111        51       3211   71         72       73
                      3111            3221       3222     91
                      111111          3311       3321     3322
                                      5111       5211     3331
                                      311111     321111   5221
                                      11111111            5311
                                                          7111
                                                          322111
                                                          331111
                                                          511111
                                                          31111111
                                                          1111111111
Also the number of integer partitions of n into an odd number of parts, the greatest of which is even. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot, A = 10) are:
  2   .  4     221   6       421     8         432       A
         211         222     22111   422       441       433
                     411             431       621       442
                     21111           611       22221     622
                                     22211     42111     631
                                     41111     2211111   811
                                     2111111             22222
                                                         42211
                                                         43111
                                                         61111
                                                         2221111
                                                         4111111
                                                         211111111
(End)

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of ranking sequences are in parentheses below.
The even-rank version is A101708 (A340605).
The even- but not necessarily positive-rank version is A340601 (A340602).
The Heinz numbers of these partitions are (A340604).
Allowing negative odd ranks gives A340692 (A340603).
- Rank -
A047993 counts balanced (rank zero) partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
- Odd -
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A058695 counts partitions of odd numbers (A300063).
A339890 counts factorizations of odd length.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A000041, A027187, A101709, A101199, A101200, A117409, A200750.

b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1)/2:
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 29 2021

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&OddQ[Max[#]]&]],{n,0,30}] (* Gus Wiseman, Feb 10 2021 *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],
     If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -
     If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1]/2;
a /@ Range[0, 55] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)

a(n) = (A000041(n) - A000025(n))/4. - Vladeta Jovovic, Dec 14 2004
G.f.: Sum((-1)^(k+1)*x^((3*k^2+k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004
a(n) = A340692(n)/2. - Gus Wiseman, Feb 07 2021

More terms from Joerg Arndt, Oct 07 2012

a(0)=0 prepended by Alois P. Heinz, Jan 29 2021

A053250 Coefficients of the '3rd-order' mock theta function phi(q).

Original entry on oeis.org

1, 1, 0, -1, 1, 1, -1, -1, 0, 2, 0, -2, 1, 1, -1, -2, 1, 3, -1, -2, 1, 2, -2, -3, 1, 4, 0, -4, 2, 3, -2, -4, 1, 5, -2, -5, 3, 5, -3, -5, 2, 7, -2, -7, 3, 6, -4, -8, 3, 9, -2, -9, 5, 9, -5, -10, 3, 12, -4, -12, 5, 11, -6, -13, 6, 16, -6, -15, 7, 15, -8, -17, 7, 19, -6, -20, 9, 19, -10, -22, 8, 25, -9, -25, 12, 25, -12, -27, 11, 31

Offset: 0

Dean Hickerson, Dec 19 1999

			G.f. = 1 + x - x^3 + x^4 + x^5 - x^6 - x^7 + 2*x^9 - 2*x^11 + x^12 + x^13 - x^14 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.12), p. 58, Eq. (26.56).
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.

T. D. Noe, Table of n, a(n) for n = 0..1000
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.
A. Folsom, K. Ono and R. C. Rhoades, Ramanujan's radial limits, 2013. - From _N. J. A. Sloane_, Feb 09 2013
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053251, A053252, A053253, A053254, A053255.

Cf. A207569, A215066.

f:=n->q^(n^2)/mul((1+q^(2*i)),i=1..n); add(f(n),n=0..10);

Series[Sum[q^n^2/Product[1+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
a[ n_] := SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ -x^2, x^2, k], {k, 0, Sqrt@ n}], {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)

{a(n) = my(t); if(n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 + x^(2*k)) + O(x^(n - (k-1)^2 + 1)), 1), n))}; /* Michael Somos, Jul 16 2007 */

Consider partitions of n into distinct odd parts. a(n) = number of them for which the largest part minus twice the number of parts is == 3 (mod 4) minus the number for which it is == 1 (mod 4).
a(n) = (-1)^n*(A027358(n)-A027357(n)). - Vladeta Jovovic, Mar 12 2006
G.f.: 1 + Sum_{k>0} x^k^2 / ((1 + x^2) (1 + x^4) ... (1 + x^(2*k))).
G.f. 1 + Sum_{n >= 0} x^(2*n+1)*Product_{k = 1..n} (x^(2*k-1) - 1) (Folsom et al.). Cf. A207569 and A215066. - Peter Bala, May 16 2017

A053253 Coefficients of the '3rd-order' mock theta function omega(q).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 14, 18, 22, 29, 36, 44, 56, 68, 82, 101, 122, 146, 176, 210, 248, 296, 350, 410, 484, 566, 660, 772, 896, 1038, 1204, 1391, 1602, 1846, 2120, 2428, 2784, 3182, 3628, 4138, 4708, 5347, 6072, 6880, 7784, 8804, 9940, 11208, 12630

Offset: 0

Dean Hickerson, Dec 19 1999

Empirical: a(n) is the number of integer partitions mu of 2n+1 such that the diagram of mu has an odd number of cells in each row and in each column. - John M. Campbell, Apr 24 2020
From Gus Wiseman, Jun 26 2022: (Start)
By Campbell's conjecture above that a(n) is the number of partitions of 2n+1 with all odd parts and all odd conjugate parts, the a(0) = 1 through a(5) = 8 partitions are (B = 11):
  (1)  (3)    (5)      (7)        (9)          (B)
       (111)  (311)    (511)      (333)        (533)
              (11111)  (31111)    (711)        (911)
                       (1111111)  (51111)      (33311)
                                  (3111111)    (71111)
                                  (111111111)  (5111111)
                                               (311111111)
                                               (11111111111)
These partitions are ranked by A352143. (End)

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 15, 17, 31.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Steven Charlton, Explicit linear dependence congruence relations for the partition function modulo 4, arXiv:2412.17459 [math.NT], 2024. See p. 3.
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053254, A053255, A261401.
Cf. A095913(n)=a(n-3).
Cf. A259094.
Conjectured to count the partitions ranked by A352143.
A069911 = strict partitions w/ all odd parts, ranked by A258116.
A078408 = partitions w/ all odd parts, ranked by A066208.
A117958 = partitions w/ all odd parts and multiplicities, ranked by A352142.
Cf. A000009, A000290, A000701, A035363 (complement A086543), A035444, A035457, A045931, A055922, A258117.

Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}]^2, {n, 0, 6}], {q, 0, 100}]

{a(n)=local(A); if(n<0, 0, A=1+x*O(x^n); polcoeff( sum(k=0, (sqrtint(2*n+1)-1)\2, A*=(x^(4*k)/(1-x^(2*k+1))^2 +x*O(x^(n-2*(k^2-k))))), n))} /* Michael Somos, Aug 18 2006 */

{a(n)=local(A); if(n<0, 0, n++; A=1+x*O(x^n); polcoeff( sum(k=0, n-1, A*=(x/(1-x^(2*k+1)) +x*O(x^(n-k)))), n))} /* Michael Somos, Aug 18 2006 */

G.f.: omega(q) = Sum_{n>=0} q^(2*n*(n+1))/((1-q)*(1-q^3)*...*(1-q^(2*n+1)))^2.
G.f.: Sum_{k>=0} x^k/((1-x)(1-x^3)...(1-x^(2k+1))). - Michael Somos, Aug 18 2006
G.f.: (1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(2*k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 10 2019
Conjectural g.f.: 1/(1 - x)*( 1 + Sum_{n >= 0} x^(3*n+1) /((1 - x)*(1 - x^3)*...*(1 - x^(2*n+1))) ). - Peter Bala, Nov 18 2024

A003406 Expansion of Ramanujan's function R(x) = 1 + Sum_{n >= 1} { x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n)) }.

Original entry on oeis.org

1, 1, -1, 2, -2, 1, 0, 1, -2, 0, 2, 0, -1, -2, 2, 1, 0, -2, 2, -2, 0, 0, 3, 0, -2, -2, 1, 0, 2, 0, 0, 0, -2, 0, 0, 1, 0, 0, 0, 2, -1, 0, -2, -2, 0, 4, 0, 2, -2, 0, -2, -1, 2, 0, -2, 2, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 4, 2, -1, 0, 0, -2, -2, -2, 2, 1, 2, 0, 0, 0, 0, -2, 2, 0, 0, -2, 2, -2, -2, 0, 3, 0, 0, 2, 0, 0, 0, -2, 1, -2, 0, -2, 0

Offset: 0

N. J. A. Sloane

a(n) = A117192(n) - A117193(n) for n>0 (number of partitions into distinct parts with even rank minus those with odd rank); see also A000025. - Reinhard Zumkeller, Mar 03 2006

Ramanujan showed that R(x) = 2*Sum_{n>=0} (S(x) - P(n,x)) - 2*S(x)*D(x), where P(n,x) = Product_{k=1..n} (1+x^k), S(x) = g.f. A000009 = P(oo,x) and D(x) = -1/2 + Sum_{n>=1} x^n/(1-x^n) = -1/2 + g.f. A000005. - Michael Somos

			1 + x - x^2 + 2*x^3 - 2*x^4 + x^5 + x^7 - 2*x^8 + 2*x^10 - x^12 - 2*x^13 + ...
q + q^25 - q^49 + 2*q^73 - 2*q^97 + q^121 + q^169 - 2*q^193 + 2*q^241 - ...

G. E. Andrews, Ramanujan's "lost" notebook V: Euler's partition identity, Adv. in Math. 61 (1986), no. 2, 156-164; Math. Rev. 87i:11137. [ The expansion in (2.8) is incorrect. ]
F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
F. J. Dyson, Selected Papers, Am. Math. Soc., 1996, p. 200.
B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989. see page 182. MR1019331 (90k:11050)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)
G. E. Andrews, Questions and conjectures in partition theory, Amer. Math. Monthly, 93 (1986), 708-711.
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000.
G. E. Andrews, F. J. Dyson and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988) 391-407.
S.-Y. Kang, Generalizations of Ramanujan's reciprocity theorem and their applications, J. London Math. Soc., 75 (2007), 18-34.
Alexander E. Patkowski, A note on the rank parity function, Discrete Math. 310 (2010), 961-965.
D. Zagier, Quantum modular forms, Example 1 in Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675

Cf. A003475, A005895, A005896, A158690.

g:=1+sum(x^(n*(n+1)/2)/product(1+x^j,j=1..n),n=1..20): gser:=series(g,x=0,110): seq(coeff(gser,x,n),n=0..104); # Emeric Deutsch, Mar 30 2006
t1:= add( (-1)^n*q^(n*(3*n+1)/2)*(1-q^(2*n+1))* add( (-1)^j*q^(-j^2),j=-n..n), n=0..20); t2:=series(t1,q,40); # N. J. A. Sloane, Jun 27 2011

max = 105; f[x_] := 1 + Sum[ x^(n*(n+1)/2) / Product[ 1+x^j, {j, 1, n}], {n, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 02 2011 *)
max = 105; s = 1 + Sum[2*q^(n*(n+1)/2)/QPochhammer[-1, q, n+1], {n, 1, Ceiling[Sqrt[2 max]]}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n) = local(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum( k=1, n, t *= if( k>1, x^k - x, x) + O(x^(n-k+2)), 1), n))} /* Michael Somos, Mar 07 2006 */

{a(n) = local(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum( k=1, (sqrtint(8*n + 1)-1)\2, t *= x^k / (1 + x^k) + x * O(x^(n - (k^2-k)/2)), 1), n))} /* Michael Somos, Aug 17 2006 */

{a(n) = local(A, p, e, x, y); if( n<0, 0, n = 24*n+1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p<5, 0, if( p%24>1 && p%24<23, if(e%2, 0, if( p%24==7 || p%24==17, (-1)^(e/2), 1)), x=y=0; if( p%24==1, forstep(i=1, sqrtint(p), 2, if( issquare( (i^2+p)/2, &y), x=i; break)), for( i=1, sqrtint(p\2), if( issquare(2*i^2 + p, &x), y=i; break))); (e+1)*(-1)^( (x + if((x-y)%6, y, -y))/6*e))))))} /* Michael Somos, Aug 17 2006 */

G.f.: 1 - Sum_{n > 0} (-x)^n * (1 - x) * (1 - x^2) * ... * (1  -x^(n-1)).
G.f.: 1 + Sum_{n>=1}(x^(n(n+1)/2)/Product_{j=1..n}(1+x^j)). - Emeric Deutsch, Mar 30 2006
Define c(24*k + 1) = A003406(k), c(24*k - 1) = -2*A003475(k), c(n) = 0 otherwise. Then c(n) is multiplicative with c(2^e) = c(3^e) = 0^e, c(p^e) = (-1)^(e/2) * (1+(-1)^e)/2 if p == 7, 17 (mod 24), c(p^e) = (1+(-1)^e)/2 if p == 5, 11, 13, 19 (mod 24), c(p^e) = (e+1)*(-1)^(y*e) where p == 1, 23 (mod 24) and p = x^2 - 72*y^2 . - Michael Somos, Aug 17 2006
Also R(x) = -2 + Sum_{n>=0} (n+1)*x^(n(n-1)/2)/(Product_{k=1..n} (1+x^k)). - Paul D. Hanna, May 22 2010

A053254 Coefficients of the '3rd-order' mock theta function nu(q).

Original entry on oeis.org

1, -1, 2, -2, 2, -3, 4, -4, 5, -6, 6, -8, 10, -10, 12, -14, 15, -18, 20, -22, 26, -29, 32, -36, 40, -44, 50, -56, 60, -68, 76, -82, 92, -101, 110, -122, 134, -146, 160, -176, 191, -210, 230, -248, 272, -296, 320, -350, 380, -410, 446, -484, 522, -566, 612, -660, 715, -772, 830, -896, 966, -1038

Offset: 0

Dean Hickerson, Dec 19 1999

In Watson 1936 the function is denoted by upsilon(q). - Michael Somos, Jul 25 2015

			G.f. = 1 - x + 2*x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 4*x^7 + 5*x^8 + ...

George E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, (Example 6, p. 29).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053253, A053255.

Cf. A058140, A067357, A376628.

Series[Sum[q^(n(n+1))/Product[1+q^(2k+1), {k, 0, n}], {n, 0, 9}], {q, 0, 100}]

/* Continued Fraction Expansion: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 + x^(n-k+1)*(1 - x^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 09 2013

G.f.: nu(q) = Sum_{n >= 0} q^(n*(n+1))/((1+q)*(1+q^3)*...*(1+q^(2*n+1)))
(-1)^n*a(n) = number of partitions of n in which even parts are distinct and if k occurs then so does every positive even number less than k.
G.f.: 1/(1 + x*(1-x)/(1 + x^2*(1-x^2)/(1 + x^3*(1-x^3)/(1 + x^4*(1-x^4)/(1 + x^5*(1-x^5)/(1 + ...)))))), a continued fraction. - Paul D. Hanna, Jul 09 2013
a(2*n) = A085140(n). a(2*n + 1) = - A053253(n). - Michael Somos, Jul 25 2015
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jun 15 2019
From Peter Bala, Jan 03 2025: (Start)
a(n) = (-1)^n * A067357(n).
nu(-q) = Sum_{n >= 0} q^n * (1 + q)*(1 + q^3)*...*(1 + q^(2*n-1)) (Andrews, p. 29: in  Example 6 take x = q and y = -q).
Conjecture: a(n) = (-1)^n * (A376628(n) + A376628(n+1)), or equivalently, (1 + q * nu(-q))/(1 + q) = Sum_{n >= 0} q^(n*(n+1))/((1 - q)*(1 - q^3)*...*(1 - q^(2*n-1))), the g.f. of A376628. (End)

A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 2, 5, 3, 4, 3, 1, 6, 4, 6, 6, 6, 2, 2, 7, 5, 8, 9, 11, 9, 7, 4, 3, 1, 8, 6, 10, 12, 16, 16, 18, 12, 12, 8, 6, 2, 2, 9, 7, 12, 15, 21, 23, 29, 27, 26, 23, 21, 15, 13, 7, 4, 3, 1, 10, 8, 14, 18, 26, 30, 40, 42, 48, 44, 46, 40, 40, 30, 26, 18, 14, 8, 6, 2, 2

Offset: 0

Alford Arnold, Jun 19 2003

There are A033638(n) values in the n-th row, compliant with the order of the polynomial.
In the example for n=6 detailed below, the orders of [6, k]_q are 1, 6, 9, 10, 9, 6, 1 for k = 0..6,
the maximum order 10 defining the row length.
Note that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.
A083480 is a variation illustrating a relationship with numeric partitions, A000041.
The rows are formed by the nonzero entries of the columns of A049597.
The coefficient of q^j in the Gaussian polynomial [n, m]q is the number of binary words on alphabet {0,1} of length n having m 1's and j inversions. Hence T(n, k) is the number of length n binary words with exactly k inversions. - _Geoffrey Critzer, May 14 2017
If n is even the n-th row converges to n+1, n-1, n-4, ..., 19, 13, 7, 4, 3, 1 which is A029552 reversed, and if n is odd the sequence is twice A098613. - Michael Somos, Jun 25 2017

			When viewed as an array with A033638(r) entries per row, the table begins:
. 1 ............... : 1
. 2 ............... : 2
. 3 1 ............. : 3 + q = (1) + (1+q) + (1)
. 4 2 2 ........... : 4 + 2q + 2q^2 = 1 + (1+q+q^2) + (1+q+q^2) + 1
. 5 3 4 3 1 ....... : 5 + 3q + 4q^2 + 3q^3 + q^4
. 6 4 6 6 6 2 2
. 7 5 8 9 11 9 7 4 3 1
. 8 6 10 12 16 16 18 12 12 8 6 2 2
. 9 7 12 15 21 23 29 27 26 23 21 15 13 7 4 3 1
...
The second but last row is from the sum over 7 q-polynomials coefficients:
. 1 ....... : 1 = [6,0]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,1]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,2]_q
. 1 1 2 3 3 3 3 2 1 1 ....... : 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9 = [6,3]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,4]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,5]_q
. 1 ....... : 1 = [6,6]_q

George E. Andrews, 'Theory of Partitions', 1976, page 242.

Seiichi Manyama, Rows n = 0..48, flattened
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Alexander Gruber, "The Egg:" Bizarre behavior of the roots of a family of polynomials Mathematics StackExchange Oct 04 2012
MathOverflow, Partition numbers and Gaussian binomial coefficient
Eric Weisstein, q-Binomial Coefficient, Mathworld.
Wikipedia, q-binomial
Index entries for sequences related to binary linear codes

Cf. A000025, A000034, A000041, A016116, A029552, A033638, A060546, A063746, A077028, A083479, A083480, A098613, A260460.

R:=PowerSeriesRing(Rationals(), 100);
qBinom:= func< n,k,x | n eq 0 or k eq 0 select 1 else (&*[(1-x^(n-j))/(1-x^(j+1)): j in [0..k-1]]) >;
A083906:= func< n,k | Coefficient(R!((&+[qBinom(n,k,x): k in [0..n]]) ), k) >;
[A083906(n,k): k in [0..Floor(n^2/4)], n in [0..12]]; // G. C. Greubel, Feb 13 2024

QBinomial := proc(n,m,q) local i ; factor( mul((1-q^(n-i))/(1-q^(i+1)),i=0..m-1) ) ; expand(%) ; end:
A083906 := proc(n,k) add( QBinomial(n,m,q),m=0..n ) ; coeftayl(%,q=0,k) ; end:
for n from 0 to 10 do for k from 0 to A033638(n)-1 do printf("%d,",A083906(n,k)) ; od: od: # R. J. Mathar, May 28 2009
T := proc(n, k) if n < 0 or k < 0 or k > floor(n^2/4) then return 0 fi;
if n < 2 then return n + 1 fi; 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) end:
seq(print(seq(T(n, k), k = 0..floor((n/2)^2))), n = 0..8);  # Peter Luschny, Feb 16 2024

Table[CoefficientList[Total[Table[FunctionExpand[QBinomial[n, k, q]], {k, 0, n}]],q], {n, 0, 10}] // Grid (* Geoffrey Critzer, May 14 2017 *)

{T(n, k) = polcoeff(sum(m=0, n, prod(k=0, m-1, (x^n - x^k) / (x^m - x^k))), k)}; /* Michael Somos, Jun 25 2017 */

def T(n,k): # T = A083906
    if k<0 or k> (n^2//4): return 0
    elif n<2 : return n+1
    else: return 2*T(n-1, k) - T(n-2, k) + T(n-2, k-n+1)
flatten([[T(n,k) for k in range(int(n^2//4)+1)] for n in range(13)]) # G. C. Greubel, Feb 13 2024

T(n, k) is the coefficient [q^k] of the Sum_{m=0..n} [n, m]_q over q-Binomial coefficients.
Row sums: Sum_{k=0..floor(n^2/4)} T(n,k) = 2^n.
For n >= k, T(n+1,k) = T(n, k) + A000041(k). - Geoffrey Critzer, Feb 12 2021
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A060546(n). - G. C. Greubel, Feb 13 2024
From Mikhail Kurkov, Feb 14 2024: (Start)
T(n, k) = 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) for n >= 2 and 0 <= k <= floor(n^2/4).
Sum_{i=0..n} T(n-i, i) = A000041(n+1). Note that upper limit of the summation can be reduced to A083479(n) = (n+2) - ceiling(sqrt(4*n)).
Both results were proved (see MathOverflow link for details). (End)
From G. C. Greubel, Feb 17 2024: (Start)
T(n, floor(n^2/4)) = A000034(n).
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A016116(n+1).
Sum_{k=0..(n + 2) - ceiling(sqrt(4*n))} (-1)^k*T(n - k, k) = (-1)^n*A000025(n+1) = -A260460(n+1). (End)

Edited by R. J. Mathar, May 28 2009

New name using a comment from Geoffrey Critzer by Peter Luschny, Feb 17 2024

A053252 Coefficients of the '3rd-order' mock theta function chi(q).

Original entry on oeis.org

1, 1, 1, 0, 0, 0, 1, 1, 0, 0, -1, 0, 1, 1, 1, -1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 1, 1, 0, -1, -1, 1, 1, 0, -1, -1, 0, 1, 2, 1, -1, -1, 0, 1, 1, 0, -1, -2, 0, 1, 2, 1, -1, -1, -1, 1, 2, 1, -1, -2, -1, 2, 2, 1, -1, -2, -1, 1, 2, 0, -1, -3, 0, 2, 3, 2, -2, -2, -1, 2, 3, 0, -2, -3, -1, 2, 3, 2, -3, -3, -1, 2, 4, 1, -2, -4, -1, 3, 4, 2, -2, -4

Offset: 0

Dean Hickerson, Dec 19 1999

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.14).
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053253, A053254, A053255, A261401.

Series[Sum[q^n^2/Product[1-q^k+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]

G.f.: chi(q) = Sum_{n >= 0} q^n^2/((1-q+q^2)*(1-q^2+q^4)*...*(1-q^n+q^(2n))).

G.f.: G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)) / G(k+1). - Joerg Arndt, Jun 29 2013

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A053251 Coefficients of the '3rd-order' mock theta function psi(q).

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A040051 Parity of partition function A000041.

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A063995 Irregular triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k.

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A101707 Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).

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A053250 Coefficients of the '3rd-order' mock theta function phi(q).

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A053253 Coefficients of the '3rd-order' mock theta function omega(q).

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