A053260 -id:A053260 - cp's OEIS frontend

A053263 Coefficients of the '5th-order' mock theta function chi_1(q).

1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 8, 9, 9, 12, 12, 15, 15, 18, 19, 23, 23, 27, 30, 33, 34, 41, 42, 49, 51, 57, 61, 69, 72, 81, 87, 96, 100, 113, 119, 132, 140, 153, 163, 180, 188, 208, 221, 240, 253, 278, 294, 319, 339, 366, 388, 422, 443, 481, 510, 549, 580, 626, 662

Offset: 0

Dean Hickerson, Dec 19 1999

The rank of a partition is its largest part minus the number of parts.
Number of partitions of n such that 2*(least part) > greatest part. - Clark Kimberling, Feb 16 2014
Also the number of partitions of n with the same median as maximum. These are conjugate to the partitions described above. For minimum instead of maximum we have A361860. - Gus Wiseman, Apr 23 2023

			From _Gus Wiseman_, Apr 20 2023: (Start)
The a(1) = 1 through a(8) = 6 partitions such that 2*(minimum) > (maximum):
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (1111)  (11111)  (222)     (322)      (53)
                                     (111111)  (1111111)  (332)
                                                          (2222)
                                                          (11111111)
The a(1) = 1 through a(8) = 6 partitions such that (median) = (maximum):
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (331)      (44)
                    (1111)  (11111)  (222)     (2221)     (332)
                                     (111111)  (1111111)  (2222)
                                                          (22211)
                                                          (11111111)
(End)

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 25

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Seiichi Manyama)
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, A053259, A053260, A053261, A053262, A053264, A053265, A053266, A053267.
Cf. A047993, A240874, A361800, A361849 (ranks A361856), A361860.
A000041 counts integer partitions, strict A000009, odd-length A027193.
A359893 and A359901 count partitions by median.
Cf. A013580, A079309, A237753, A237757, A237800, A361848, A361858, A361859.

1+Series[Sum[q^(2n+1)(1+q^n)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
(* Also: *)
Table[Count[ IntegerPartitions[n], p_ /; 2 Min[p] > Max[p]], {n, 40}]
(* Clark Kimberling, Feb 16 2014 *)
nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)*(1+x^k) / Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[nmax/2]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

G.f.: chi_1(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
G.f.: chi_1(q) = 1 + Sum_{n>=0} q^(2n+1) (1+q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
a(n) is twice the number of partitions of 5n+3 with rank == 2 (mod 5) minus number with rank == 0 or 1 (mod 5).
a(n) - 1 is the number of partitions of n with unique smallest part and all other parts <= one plus twice the smallest part.
a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 16 2019

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

Original entry on oeis.org

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

Dean Hickerson, Dec 19 1999

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]

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.

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.

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

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

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 *)

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 */

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

A053258 Coefficients of the '5th-order' mock theta function phi_0(q).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 4, 4, 5, 4, 4, 5, 5, 5, 5, 6, 6, 5, 5, 6, 6, 6, 6, 7, 7, 7, 6, 7, 8, 7, 8, 8, 9, 9, 8, 9, 10, 9, 9, 10, 11, 10, 10, 11, 11, 11, 11, 12

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, 23, 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, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Series[Sum[q^n^2 Product[1+q^(2k-1), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

G.f.: phi_0(q) = Sum_{n>=0} q^n^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 and if k occurs as a part then all smaller positive odd numbers occur.
a(n) ~ sqrt(phi) * exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053266 Coefficients of the '5th-order' mock theta function Phi(q).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 9, 10, 12, 12, 14, 15, 17, 18, 20, 21, 25, 26, 29, 31, 35, 36, 41, 43, 48, 51, 56, 59, 66, 70, 76, 81, 89, 94, 103, 109, 119, 126, 137, 144, 158, 167, 180, 191, 207, 218, 236, 250, 269, 285, 306, 323, 349, 368

Offset: 0

Dean Hickerson, Dec 19 1999

In Ramanujan's lost notebook the generating function is denoted by phi(q) on pages 18 and 20, however on page 18 there is no minus one first term. - Michael Somos, Jul 07 2015

			G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, MR2952081, See p. 12, Equation (2.1.18) and also page 26 equation (2.4.8).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053267.

Cf. A259910.

a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24 n/5]}, SeriesCoefficient[ -1 + Sum[ (-1)^k x^(5 k (3 k + 1)/2) / (1 - x^(5 k + 1)), {k, Quotient[m + 1, -6], Quotient[m - 1, 6]}] / QPochhammer[ x^5], {x, 0, n}]]]; (* Michael Somos, Jul 07 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n\5), x^(5*k^2) / prod(i=1, 5*k+1, 1 - if( i%5==1 || i%5==4, x^i), 1 + x * O(x^(n - 5*k^2)))) - 1, n))}; /* Michael Somos, Jul 07 2015 */

{a(n) = my(A, m); if( n<0, 0, m = sqrtint(1 + 24*n\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, (-1)^k * x^(5*k*(3*k + 1)/2) / (1 - x^(5*k + 1)), A) / eta(x^5 + A) - 1, n))}; /* Michael Somos, Jul 07 2015 */

G.f.: -1 + Sum_{k>=0} q^(5k^2)/((1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^(5k+1))).
3*a(n) = A053262(n) + A259910(n) unless n=0. - Michael Somos, Jul 07 2015
a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(3/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053256 Coefficients of the '5th-order' mock theta function f_0(q).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, -1, 2, -2, 2, -1, 1, -3, 2, -1, 3, -3, 2, -2, 3, -4, 3, -3, 4, -5, 5, -3, 5, -7, 5, -5, 6, -7, 7, -6, 7, -9, 9, -7, 9, -11, 9, -9, 11, -13, 12, -11, 13, -15, 15, -13, 16, -19, 17, -17, 19, -21, 21, -20, 22, -26, 25, -23, 27, -30, 29, -28, 32, -35, 34, -34, 36, -41, 40, -38, 44, -48, 46

Offset: 0

Dean Hickerson, Dec 19 1999

In Ramanujan's lost notebook page 21 is written the g.f. neatly crossed out between the 3rd and 4th equations. - Michael Somos, Feb 13 2017

			G.f. = 1 + x - x^2 + x^3 - x^6 + x^7 + x^9 - 2*x^10 + x^11 - x^12 + 2*x^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, pp. 19, 21, 22, 23.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
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.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.
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 A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

N:= 100: # for a(0)..a(N)
g:= add(q^(k^2)/mul(1+q^i,i=1..k),k=0..floor(sqrt(N))):
S:= series(g,q,N+1):
seq(coeff(S,q,k),k=0..N)]; # Robert Israel, Mar 27 2018

Series[Sum[q^n^2/Product[1+q^k, {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
a[ n_] := SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ -x, x, k] // FunctionExpand, {k, 0, Sqrt@ n}], {x, 0, n}]; (* Michael Somos, Feb 13 2017 *)

{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^k + O(x^(n - (k-1)^2 + 1))), 1), n))}; /* Michael Somos, Mar 12 2006 */

G.f.: 1 + Sum_{k>0} q^k^2 / ((1 + q) * (1 + q^2) * ... * (1 + q^k)).
Consider partitions of n into parts differing by at least 2. For n > 0: a(n) is the number of them with largest part odd minus number with largest part even.
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 15 2019

A053262 Coefficients of the 5th-order mock theta function chi_0(q).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 7, 9, 7, 12, 11, 13, 13, 17, 15, 21, 20, 24, 24, 29, 28, 36, 35, 40, 42, 50, 48, 58, 58, 67, 70, 80, 79, 93, 95, 106, 111, 125, 127, 145, 149, 166, 172, 191, 196, 222, 229, 250, 262, 289, 298, 330, 343, 373, 391, 427, 442, 486

Offset: 0

Dean Hickerson, Dec 19 1999

The rank of a partition is its largest part minus the number of parts.

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. 20, 23, 25.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (corrected and extended previous b-file 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, A053259, A053260, A053261, A053263, A053264, A053265, A053266, A053267.

1+Series[Sum[q^(2n+1)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[nmax/2]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

G.f.: chi_0(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))).
G.f.: chi_0(q) = 1 + Sum_{n>=0} q^(2n+1)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
a(n) is the number of partitions of 5n with rank == 1 (mod 5) minus number with rank == 0 (mod 5).
a(n) is the number of partitions of n with unique smallest part and all other parts <= twice the smallest part.
a(n) is the number of partitions where the largest part is odd and all other parts are greater than half of the largest part. - N. Sato, Jan 21 2010
a(n) ~ exp(Pi*sqrt(2*n/15)) / sqrt((5 + sqrt(5))*n). - Vaclav Kotesovec, Jun 12 2019

A053264 Coefficients of the '5th-order' mock theta function F_0(q).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 10, 11, 11, 13, 14, 15, 17, 18, 19, 22, 24, 25, 28, 30, 32, 36, 39, 41, 45, 49, 52, 57, 61, 65, 71, 76, 81, 88, 94, 100, 109, 116, 123, 133, 142, 151, 163, 174, 184, 198, 211, 224, 240, 255, 271, 290

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. 20, 22, 23, 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, A053259, A053260, A053261, A053262, A053263, A053265, A053266, A053267.

Series[Sum[q^(2n^2)/Product[1-q^(2k+1), {k, 0, n-1}], {n, 0, 7}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(2*k^2) / Product[1-x^(2*j+1), {j, 0, k-1}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

G.f.: F_0(q) = Sum_{n>=0} q^(2n^2)/((1-q)(1-q^3)...(1-q^(2n-1))).
a(n) is the number of partitions of n into odd parts, each of which occurs at least twice, such that if k occurs then all smaller positive odd numbers occur.
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(3/2)*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053267 Coefficients of the '5th-order' mock theta function Psi(q).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 4, 4, 5, 5, 7, 6, 8, 8, 9, 9, 12, 11, 14, 14, 16, 16, 20, 19, 23, 24, 27, 27, 32, 32, 37, 38, 43, 44, 51, 51, 58, 61, 67, 69, 78, 80, 89, 93, 102, 106, 118, 121, 134, 140, 153, 159, 175, 181, 198, 207, 224, 234, 256, 265, 288

Offset: 0

Dean Hickerson, Dec 19 1999

Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266.

Series[Sum[q^(5n^2)/Product[1-q^Abs[5k+2], {k, -n, n}], {n, 0, 4}], {q, 0, 100}]-1
nmax = 100; CoefficientList[Series[-1 + Sum[x^(5*k^2)/ Product[1-x^Abs[5*j+2], {j, -k, k}], {k, 0, Floor[Sqrt[nmax/5]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

G.f.: Psi(q) = -1 + Sum_{n>=0} q^(5n^2)/((1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^(5n+2))).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(3/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053265 Coefficients of the '5th-order' mock theta function F_1(q).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 20, 21, 24, 26, 28, 31, 34, 37, 40, 44, 47, 51, 56, 60, 65, 71, 76, 82, 89, 95, 103, 111, 119, 128, 138, 148, 158, 171, 182, 195, 210, 223, 239, 256, 273, 292, 312, 332, 354, 378, 402, 428

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. 20, 22, 25.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
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, A053259, A053260, A053261, A053262, A053263, A053264, A053266, A053267.

Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 6}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(2*k*(k+1)) / Product[1-x^(2*j+1), {j, 0, k}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

G.f.: F_1(q) = Sum_{n>=0} q^(2n(n+1))/((1-q)(1-q^3)...(1-q^(2n+1))).

a(n) ~ sqrt(phi) * exp(Pi*sqrt(2*n/15)) / (2^(3/2)*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053257 Coefficients of the '5th-order' mock theta function f_1(q).

Original entry on oeis.org

1, 0, 1, -1, 1, -1, 2, -2, 1, -1, 2, -2, 2, -2, 2, -3, 3, -2, 3, -4, 4, -4, 4, -5, 5, -4, 5, -6, 6, -6, 7, -8, 7, -7, 8, -9, 10, -9, 10, -12, 11, -11, 13, -14, 14, -15, 16, -17, 17, -16, 19, -21, 20, -21, 23, -25, 25, -25, 27, -29, 30, -30, 32, -35, 35, -35, 39, -41, 41, -43, 45, -49, 50, -49, 53, -57, 58, -59, 63, -67, 68

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.

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.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.
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, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Series[Sum[q^(n^2+n)/Product[1+q^k, {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2+k) / Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)

G.f.: f_1(q) = Sum_{n>=0} q^(n^2+n)/((1+q)(1+q^2)...(1+q^n)).
Consider partitions of n into parts differing by at least 2 and with smallest part at least 2. a(n) is the number of them with largest part even minus number with largest part odd.
a(n) ~ (-1)^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 15 2019

cp's OEIS Frontend

A053263 Coefficients of the '5th-order' mock theta function chi_1(q).

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

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A053258 Coefficients of the '5th-order' mock theta function phi_0(q).

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A053266 Coefficients of the '5th-order' mock theta function Phi(q).

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A053256 Coefficients of the '5th-order' mock theta function f_0(q).

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A053262 Coefficients of the 5th-order mock theta function chi_0(q).

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A053264 Coefficients of the '5th-order' mock theta function F_0(q).

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