A053260 -id:A053260 - cp's OEIS frontend

A053259 Coefficients of the '5th-order' mock theta function phi_1(q).

0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 4, 2, 2, 4, 4, 3, 3, 4, 4, 4, 3, 4, 5, 4, 4, 5, 5, 4, 4, 5, 6, 5, 4, 6, 7, 5, 5, 6, 7, 6, 6, 7, 7, 7, 6, 8, 9, 7, 7, 9

Offset: 0

Dean Hickerson, Dec 19 1999

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22, 25.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
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.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Series[Sum[q^(n+1)^2 Product[1+q^(2k-1), {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^((k+1)^2) * Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

G.f.: phi_1(q) = Sum_{n>=0} q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1)).
a(n) is the number of partitions of n into odd parts such that each occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive odd numbers occur.
a(n) ~ exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A227536 Irregular triangle read by rows, T(n,k) is the sum of base boxes of Pyramid arranged by n-boxes in k-th patterns.

Original entry on oeis.org

2, 3, 0, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 5, 7, 6, 6, 7, 6, 7, 6, 7, 7, 8, 7, 6, 8, 7, 7, 8, 7, 8, 8, 7, 9, 8, 7, 8, 8, 7, 9, 8, 8, 9, 8, 8, 7, 9, 9, 8, 8, 10, 9, 9, 9, 8, 10, 9, 9, 8, 10, 9, 9, 9, 8, 10, 10, 9, 9, 8, 10, 10, 9, 9, 9, 11, 10, 10, 9, 9, 8, 10, 10, 9, 10, 9, 9

Offset: 3

Kival Ngaokrajang, Jul 15 2013

The rules for Pyramid arrangement are: (1) boxes shall be arranged in symmetrical forms; (2) each step width shall be 0.5 or 1, where boxes width = 1.

The number of patterns on each n-th step is A053260(n).

			For n = 3..6.
   [1]    [1]    [1|2]     [1]     [1]       [1]
  [2|3] [2|3|4] [3|4|5] [2|3|4|5]  [2]      [2|3]
                                 [3|4|5]   [4|5|6]
  T(3,1) = 2, T(4,1) = 3, T(5,k) = 0 {no pattern exist due to step width vilolations i.e. [0.5,2], [1.5,1] & [1,0,1]}, T(6,1) = 3, ...
The triangle begins:
n/k 1 2 3 4 5
3   2
4   3
5   0
6   3
7   4
8   4
9   5
10  4
11  5
12  5
13  6 5
14  6
15  6 5
16  7 6
17  6
18  7 6
19  7 6
20  7 7
21  8 7 6
22  8 7 7
23  8 7
24  8 8 7
25  9 8 7
26  8 8 7
27  9 8 8
28  9 8 8 7
29  9 9 8 8
30  9 9 8 8
31  10 9 9 9 8
...
For n = 42, T(n,k) = 11, 11, 11, 10, 11, 10, 10, 9; see illustration in links.

Kival Ngaokrajang, Illustration for n = 42, k = 1..8

Cf. A053260.

cp's OEIS Frontend

A053259 Coefficients of the '5th-order' mock theta function phi_1(q).

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