A053254 -id:A053254 - 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.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

A000025 Coefficients of the 3rd-order mock theta function f(q).

Original entry on oeis.org

1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n with even rank minus number with odd rank. The rank of a partition is its largest part minus the number of parts.

			G.f. = 1 + q - 2*q^2 + 3*q^3 - 3*q^4 + 3*q^5 - 5*q^6 + 7*q^7 - 6*q^8 + 6*q^9 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 82, Examples 4 and 5.
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe and then corrected by Sean A. Irvine, Apr 25 2019)
George E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 95.
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. [See f(q)]
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11), (26.24).
Ken Ono, The last words of a genius, Notices Amer. math. Soc., 57 (2010), 1410-1419.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.
Eric Weisstein's World of Mathematics, Mock Theta Function.

Other '3rd-order' mock theta functions are at A013953, A053250, A053251, A053252, A053253, A053254, A053255. See also A000039, A000199.

Cf. A340601, A340692.

a:= m-> coeff(series((1+4*add((-1)^n*q^(n*(3*n+1)/2)/
        (1+q^n), n=1..m))/mul(1-q^i, i=1..m), q, m+1), q, m):
seq(a(n), n=0..120);

CoefficientList[Series[(1+4Sum[(-1)^n q^(n(3n+1)/2)/(1+q^n), {n, 1, 10}])/Sum[(-1)^n q^(n(3n+1)/2), {n, -8, 8}], {q, 0, 100}], q] (* N. J. A. Sloane *)
sgn[P_ (* a partition *)] :=
Signature[
  PermutationList[
   Cycles[Flatten[
     SplitBy[Range[Total[P]], (Function[{x}, x > #1] &) /@
       Accumulate[P]], Length[P] - 1]]]]
conjugate[P_List(* a partition *)] :=
Module[{s = Select[P, #1 > 0 &], i, row, r}, row = Length[s];
  Table[r = row; While[s[[row]] <= i, row--]; r, {i, First[s]}]]
Total[Function[{x}, sgn[x] sgn[conjugate[x]]] /@
    IntegerPartitions[#]] & /@ Range[20]
(* George Beck, Oct 25 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / Product[ 1 + x^j, {j, k}]^2, {k, 0, Sqrt@n}], {x, 0, n}]]; (* Michael Somos, Jun 30 2015 *)
rnk[prts_]:=Max[prts]-Length[prts]; mtf[n_]:=Module[{pn=IntegerPartitions[n]},Total[If[ EvenQ[ rnk[#]],1,-1]&/@pn]]; Join[{1},Array[mtf,60]] (* Harvey P. Dale, Sep 13 2024 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(i=1, k, 1 + x^i, 1 + x * O(x^(n - k^2)))^2, 1), n))}; /* Michael Somos, Sep 02 2007 */

my(N=60, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)^2/(1+x^k))) \\ Seiichi Manyama, May 23 2023

G.f.: 1 + Sum_{n>=1} (q^(n^2) / Product_{i=1..n} (1 + q^i)^2).
G.f.: (1 + 4 * Sum_{n>=1} (-1)^n * q^(n*(3*n+1)/2) / (1 + q^n)) / Product_{i>=1} (1 - q^i).
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)) [Ramanujan]. - Vaclav Kotesovec, Jun 10 2019
G.f.: 1 - Sum_{n >= 1} (-1)^n*x^n/Product_{k = 1..n} 1 + x^k. See Fine, equation 26.22, p. 55. - Peter Bala, Feb 04 2021
From Seiichi Manyama, May 23 2023: (Start)
a(n) = A340601(n) - A340692(n).
G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k)^2 / (1+x^k). (End)

Entry improved by comments from Dean Hickerson

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

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

A053255 Coefficients of the '3rd-order' mock theta function rho(q).

Original entry on oeis.org

1, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, 0, 2, -1, -1, 1, -1, -1, 2, -1, 0, 2, -1, -1, 2, -2, -1, 3, -2, -1, 3, -2, -1, 3, -2, -1, 4, -3, -1, 4, -2, -2, 4, -3, -2, 5, -4, -2, 6, -3, -2, 6, -4, -2, 7, -5, -2, 7, -5, -3, 8, -6, -3, 9, -6, -3, 10, -6, -4, 10, -7, -4, 12, -8, -4, 13, -8, -5, 13, -9, -5, 15, -10, -5, 16, -11, -6, 17, -12, -7, 19, -13, -6, 21, -13

Offset: 0

Dean Hickerson, Dec 19 1999

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

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, A053252, A053253, A053254.

Series[Sum[q^(2n(n+1))/Product[1+q^(2k+1)+q^(4k+2), {k, 0, n}], {n, 0, 6}], {q, 0, 100}]

G.f.: rho(q) = Sum_{n >= 0} q^(2*n*(n+1))/((1+q+q^2)*(1+q^3+q^6)*...*(1+q^(2*n+1)+q^(4*n+2))).

A085140 Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 10, 12, 15, 20, 26, 32, 40, 50, 60, 76, 92, 110, 134, 160, 191, 230, 272, 320, 380, 446, 522, 612, 715, 830, 966, 1120, 1292, 1494, 1720, 1976, 2272, 2602, 2974, 3400, 3876, 4412, 5020, 5700, 6460, 7322, 8282, 9352, 10559, 11900, 13396

Offset: 0

Michael Somos, Jun 20 2003

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In the notation of Dragonette on page 498 Lemma 6, the generating function is G_2(q^(1/2))/2.
Equals A000009 convolved with A010054. [Gary W. Adamson, Mar 16 2010]

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 15*x^8 + ...
G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 12*q^43 + 15*q^49 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.
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. 16.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A010054. [Gary W. Adamson, Mar 16 2010]

Cf. A053254, A132970.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k) * (1 + x^k)^3, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 2, n, 2}] / Product[ 1 - x^k, {k, 1, n, 2}]^2, {x, 0, n}];
a[ n_] := With[ {t = Log[q]/(2 Pi I)}, SeriesCoefficient[ q^(-1/6) DedekindEta[ 2 t]^3 / DedekindEta[ t]^2, {q, 0, n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x, x]^3, {x, 0, n}]; (* Michael Somos, Jul 11 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(j j + j) / Product[ 1 + x^k, {k, 1, 2 j + 1, 2}], {j, 0, Sqrt[8 n + 1]/2}], {x, 0, 2 n}]]; (* Michael Somos, Jul 11 2015 *)

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

Expansion of psi(x) / chi(-x) = f(-x^2) / chi(-x)^2 = f(-x) / chi(-x)^3 = phi(-x) / chi(-x)^4 = phi(x) / chi(-x^2)^2 = f(-x^2)^2 / phi(-x) = f(-x)^4 / phi(-x)^3 = psi(x)^2 / f(-x^2) = chi(x)^2 * psi(x^2) = f(-x^2)^3 / f(-x)^2 in powers of x where f(), phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Feb 18 2006
Euler transform of period 2 sequence [ 2, -1, ...].
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1))^2 = Product_{k>0} (1 - x^k) * (1 + x^k)^3.
a(n) = b(n)+b(n-1)+b(n-3)+b(n-6)+...+b(n-k*(k+1)/2)+..., where b() is A000009(). E.g. a(8) = b(8)+b(7)+b(5)+b(2) = 6+5+3+1 = 15. - Vladeta Jovovic, Aug 18 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (3/4)^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132970. - Michael Somos, Jul 11 2015
a(n) = A053254(2*n). - Michael Somos, Jul 11 2015
a(n) ~ exp(Pi*sqrt(n/3))/(4*sqrt(n)). - Vaclav Kotesovec, Sep 07 2015

A067357 Number of self-conjugate partitions of 4n+1 into odd parts.

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, 1120

Offset: 0

Naohiro Nomoto, Feb 24 2002

Also 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 (Dean Hickerson). Absolute values of the terms of A053254. - Emeric Deutsch, Feb 10 2006
The number of self-conjugate partitions of n into odd parts is nonzero if and only if n = 4*k + 1 for some nonnegative integer k. - Michael Somos, Jul 25 2015
Also number of C3v plane partitions of n = 3*k + 1 with rank 1 ; equivalently number of self-conjugate integer partitions with (weight-length) = n. - Wouter Meeussen, May 23 2025

			a(5)=3 because we have [11,1,1,1,1,1,1,1,1,1,1], [9,3,3,1,1,1,1,1,1] and [5,5,5,3,3].
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 + ...

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 260, Article 512.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. S. Andersen, A Bijection Between Two Different Classes of Partitions Enumerated by p_nu(n), arXiv:2007.09794 [math.CO], 2020.
George E. Andrews, The Bhargava-Adiga Summation and Partitions, Journal of the Indian Mathematical Society, Volume 84, Issue 3-4, 2017.
George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, to appear in Annals of Combinatorics, 2017.

Cf. A000700, A000701.

Cf. A053253, A053254, A085140, A047993.

g:=sum(q^(k*(k+1))/product(1-q^(2*j+1),j=0..k),k=0..8): gser:=series(g,q=0,80): seq(coeff(gser,q,n),n=0..75); # Emeric Deutsch, Feb 10 2006

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / Product[ 1 - x^i, {i, 1, 2 k + 1, 2}], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* Michael Somos, Jul 25 2015 *)
Table[Length[Flatten[Table[Select[IntegerPartitions[w], (w-Length[#])== r && TransposePartition[#] == # &],{w,r,1+2r}],1]],{r,1,17}] (* Wouter Meeussen, May 24 2025 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 4*n+1) -1) \ 2, x^(k^2 + k) / prod(j=0, k, 1 - x^(2*j+1), 1 + x * O(x^(n - k^2 - k)))), n))}; /* Michael Somos, Jan 27 2008 */

/* 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.: Sum_{k>=0} q^(k*(k+1)) / ((1-q) * (1-q^3) ... (1-q^(2*k+1))). - Emeric Deutsch and Dean Hickerson
G.f.: Sum_{k>=0} q^k * (1+q) * (1+q^3) ... (1+q^(2*k-1)). - Dean Hickerson and Vladeta Jovovic
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
From Michael Somos, Jul 25 2015: (Start)
Expansion of nu(-x) in powers of x where nu() is a 3rd-order mock theta function.
a(n) = (-1)^n * A053254(n).
a(2*n) = A085140(n).
a(2*n + 1) = A053253(n). (End)
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jun 15 2019

More terms from Emeric Deutsch, Feb 10 2006

A376628 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1)).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 10, 10, 12, 14, 15, 17, 19, 21, 23, 27, 29, 31, 37, 39, 43, 49, 52, 58, 64, 70, 76, 84, 92, 99, 111, 119, 129, 143, 153, 167, 183, 197, 213, 233, 251, 271, 295, 317, 343, 372, 400, 430, 466, 500, 538, 582, 622, 670

Offset: 0

Vaclav Kotesovec, Sep 30 2024

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

Cf. A003106, A053251, A053254, A053258, A053282, A333179.

nmax=100; CoefficientList[Series[Sum[x^(k*(k+1))/Product[1-x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} Product_{j=1..k} x^(2*j)/(1 - x^(2*j-1)).
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/2) * sqrt(n)).
Conjectural g.f.: (1 + q * nu(-q))/(1 + q) = 1 + Sum_{k >= 0} q^(k+2)*Product_{j = 1..k} 1 + q^(2*j+1), where nu(q) is the g.f. of A053254. - Peter Bala, Jan 03 2025

A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

Original entry on oeis.org

1, 3, 6, 11, 17, 27, 38, 55, 76, 103, 136, 182, 235, 303, 385, 489, 612, 766, 945, 1166, 1428, 1742, 2111, 2557, 3072, 3686, 4401, 5246, 6223, 7371, 8692, 10236, 12014, 14074, 16435, 19171, 22292, 25884, 29981, 34677, 40017, 46122, 53038, 60920

Offset: 0

Jonathan Vos Post, Aug 28 2005

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000039, A000019, A053250, A053251, A053252, A053253, A053254, A053255, A064053.

nmax = 200; f[q_, s_] := Sum[q^(n^2)/Product[1 + q^k, {k, n}]^2, {n, 0, s}]; A000039:= CoefficientList[Series[f[q, nmax], {q, 0, nmax}], q][[1 ;; -1 ;; 2]]; Table[Sum[Abs[A000039[[k]]], {k,1,n}], {n,1,51}] (* G. C. Greubel, Feb 18 2018 *)

a(n) = Sum_{k=0..n} abs(A000039(k)). [corrected by Joerg Arndt, Feb 25 2018]

a(n) ~ sqrt(3/2) * exp(sqrt(n/3)*Pi) / Pi. - Vaclav Kotesovec, Jun 12 2019

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A000025 Coefficients of the 3rd-order mock theta function f(q).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A053255 Coefficients of the '3rd-order' mock theta function rho(q).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A085140 Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.

A376628 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1)).