A237800 -id:A237800 - 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 25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

A118084 Number of partitions of n such that largest part k occurs at most floor(k/2) times.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 11, 16, 23, 33, 46, 63, 86, 116, 153, 203, 265, 345, 444, 571, 727, 925, 1166, 1468, 1836, 2293, 2845, 3525, 4345, 5347, 6550, 8011, 9758, 11867, 14380, 17399, 20984, 25269, 30341, 36376, 43500, 51943, 61877, 73608, 87373, 103571

Offset: 1

Emeric Deutsch, Apr 12 2006

nonn

Also number of partitions of n such that if the number of parts is k, then the smallest part is at most floor(k/2). Example: a(8)=16 because we have [7,1],[6,1,1],[5,2,1],[4,3,1],[5,1,1,1],[4,2,1,1],[3,3,1,1],[3,2,2,1],[2,2,2,2],[4,1,1,1,1],[3,2,1,1,1],[2,2,2,1,1],[3,1,1,1,1,1],[2,2,1,1,1,1],[2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1].

			a(8)=16 because we have [8],[7,1],[6,2],[6,1,1],[5,3],[5,2,1],[5,1,1,1],[4,4],[4,3,1],[4,2,2],[4,2,1,1],[4,1,1,1,1],[3,2,2,1],[3,2,1,1,1],[3,1,1,1,1,1] and [2,1,1,1,1,1,1].

Cf. A118082, A118083.

Cf. A237758, A237757, A237799, A237800. - Clark Kimberling, Feb 15 2014

g:=sum(x^k*(1-x^(k*(floor(k/2))))/product(1-x^j,j=1..k),k=1..85): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..50);

z=55 ; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := Length[p];
Table[Count[q[n], p_ /; 2 Min[p] <= t[p]], {n,z}] (* Clark Kimberling, Feb 15 2014 *)

G.f.=sum(x^k*(1-x^(k(floor(k/2))))/product(1-x^j, j=1..k), k=1..infinity).

A237758 Number of partitions of n such that 2*(least part) < number of parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 10, 14, 21, 30, 42, 58, 80, 108, 144, 192, 252, 329, 426, 549, 702, 895, 1131, 1427, 1789, 2237, 2781, 3450, 4259, 5247, 6436, 7878, 9607, 11693, 14182, 17172, 20727, 24974, 30008, 35997, 43072, 51457, 61330, 72988, 86677, 102785, 121645

Offset: 1

Clark Kimberling, Feb 15 2014

			a(5) = 4 counts these partitions: 311, 221, 2111, 11111.

Cf. A118084, A237757, A237799, A237800.

z = 55; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := Length[p];
Table[Count[q[n], p_ /; 2 Min[p] < t[p]], {n, z}]   (* A237758 *)
Table[Count[q[n], p_ /; 2 Min[p] <= t[p]], {n, z}]  (* A118084 *)
Table[Count[q[n], p_ /; 2 Min[p] == t[p]], {n, z}]  (* A237757 *)
Table[Count[q[n], p_ /; 2 Min[p] > t[p]], {n, z}]   (* A237799 *)
Table[Count[q[n], p_ /; 2 Min[p] >= t[p]], {n, z}]  (* A237800 *)

A237799 Number of partitions of n such that 2*(least part) > number of parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 6, 7, 9, 10, 14, 15, 19, 23, 28, 32, 40, 46, 56, 65, 77, 89, 107, 122, 143, 165, 193, 220, 257, 292, 338, 385, 443, 503, 578, 653, 746, 844, 962, 1083, 1231, 1384, 1567, 1761, 1987, 2227, 2510, 2807, 3153, 3523, 3949, 4403, 4927, 5485

Offset: 1

Clark Kimberling, Feb 15 2014

			a(7) = 4 counts these partitions: 7, 52, 43, 322.

Cf. A237758, A118084, A237757, A237800.

z = 55; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := Length[p];
Table[Count[q[n], p_ /; 2 Min[p] < t[p]], {n, z}]   (* A237758 *)
Table[Count[q[n], p_ /; 2 Min[p] == t[p]], {n, z}]  (* A237757 *)
Table[Count[q[n], p_ /; 2 Min[p] > t[p]], {n, z}]   (* A237799 *)
Table[Count[q[n], p_ /; 2 Min[p] >= t[p]], {n, z}]  (* A237800 *)

A362049 Number of integer partitions of n such that (length) = 2*(median).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 3, 3, 3, 3, 3, 3, 4, 5, 9, 12, 19, 22, 29, 32, 39, 43, 51, 57, 70, 81, 101, 123, 153, 185, 230, 272, 328, 386, 454, 526, 617, 708, 824, 951, 1106, 1277, 1493, 1727, 2020, 2344, 2733, 3164, 3684, 4245, 4914, 5647, 6502, 7438, 8533, 9730

Offset: 1

Gus Wiseman, Apr 10 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). All of these partitions have even length, because an odd-length multiset cannot have fractional median.

			The a(13) = 3 through a(15) = 5 partitions:
  (7,2,2,2)  (8,2,2,2)      (9,2,2,2)
  (8,2,2,1)  (9,2,2,1)      (10,2,2,1)
  (8,3,1,1)  (9,3,1,1)      (10,3,1,1)
             (3,3,3,3,1,1)  (3,3,3,3,2,1)
                            (4,3,3,3,1,1)

For maximum instead of median we have A237753.
For minimum instead of median we have A237757.
For maximum instead of length we have A361849, ranks A361856.
This is the equal case of A362048.
These partitions have ranks A362050.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A013580, A027193, A079309, A237800, A240219, A361801, A361848, A361858, A361859.

Table[Length[Select[IntegerPartitions[n],Length[#]==2*Median[#]&]],{n,30}]

A362048 Number of integer partitions of n such that (length) <= 2*(median).

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 8, 12, 15, 20, 25, 33, 41, 53, 66, 85, 105, 134, 164, 205, 250, 308, 373, 456, 549, 666, 799, 963, 1152, 1382, 1645, 1965, 2330, 2767, 3269, 3865, 4546, 5353, 6274, 7357, 8596, 10046, 11700, 13632, 15834, 18394, 21312, 24690, 28534, 32974

Offset: 1

Gus Wiseman, Apr 10 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(1) = 1 through a(9) = 15 partitions:
  (1)  (2)   (3)   (4)   (5)    (6)    (7)     (8)     (9)
       (11)  (21)  (22)  (32)   (33)   (43)    (44)    (54)
                   (31)  (41)   (42)   (52)    (53)    (63)
                         (221)  (51)   (61)    (62)    (72)
                                (222)  (322)   (71)    (81)
                                (321)  (331)   (332)   (333)
                                       (421)   (422)   (432)
                                       (2221)  (431)   (441)
                                               (521)   (522)
                                               (2222)  (531)
                                               (3221)  (621)
                                               (3311)  (3222)
                                                       (3321)
                                                       (4221)
                                                       (4311)

For maximum instead of median we have A237755.
For minimum instead of median we have A237800.
For maximum instead of length we have A361848.
The equal case is A362049.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A013580, A027193, A237824, A240219, A361394, A361851, A361856-A361860, A361867, A361868.

Table[Length[Select[IntegerPartitions[n],Length[#]<=2*Median[#]&]],{n,30}]

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

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A118084 Number of partitions of n such that largest part k occurs at most floor(k/2) times.

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A237758 Number of partitions of n such that 2*(least part) < number of parts.

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A237799 Number of partitions of n such that 2*(least part) > number of parts.

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A362049 Number of integer partitions of n such that (length) = 2*(median).

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A362048 Number of integer partitions of n such that (length) <= 2*(median).

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