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

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

Author

Dean Hickerson, Dec 19 1999

Keywords

• nonn
• easy

Comments

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)

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 15, 17, 31.

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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

PARI

{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 */

PARI

{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 */

Formula

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