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

0,2

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 15, 17, 31.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

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.

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

Mathematica

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

Prog

(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 */

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.

Sequence in context: A217123 A014977 A008583 * A095913 A102848 A260183

Adjacent sequences: A053250 A053251 A053252 * A053254 A053255 A053256

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved