A053261 Coefficients of the '5th-order' mock theta function psi_1(q).

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 10, 10, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 34, 35, 37, 39, 40, 41, 44, 45, 47, 50, 51, 53, 56, 58, 60, 63, 65

Offset: 0

Author

Dean Hickerson, Dec 19 1999

Keywords

• nonn
• easy

Comments

Number of partitions of n such that each part occurs at most twice and if k occurs as a part then all smaller positive integers occur.

Strictly unimodal compositions with rising range 1, 2, 3, ..., m where m is the largest part and distinct parts in the falling range (this follows trivially from the comment above). [Joerg Arndt, Mar 26 2014]

References

• 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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
• George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
• George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
• William J. Keith, Partitions into parts simultaneously regular, distinct, and/or flat, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence.
• George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053262, A053263, A053264, A053265, A053266, A053267.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+1), j=1..min(2, n/i))))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 26 2014

Mathematica

Series[Sum[q^(n(n+1)/2) Product[1+q^k, {k, 1, n}], {n, 0, 13}], {q, 0, 100}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, Sum[b[n - i*j, i + 1], {j, 1, Min[2, n/i]}]]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 09 2018, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

PARI

N = 66;  x = 'x + O('x^N); gf = sum(n=0,N, x^(n*(n+1)/2) * prod(k=1,n,1+x^k) ); v = Vec(gf) /* Joerg Arndt, Apr 21 2013 */

Formula

G.f.: psi_1(q) = Sum_{n>=0} q^(n*(n+1)/2) * Product_{k=1..n} (1 + q^k).

a(n) ~ sqrt(phi) * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019