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Search: a237800 -id:a237800
Displaying 1-6 of 6 results found.
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A053263
Coefficients of the '5th-order' mock theta function chi_1(q).
+10
50
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 25
LINKS
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.
FORMULA
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
EXAMPLE
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)
MATHEMATICA
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 *)
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A118084
Number of partitions of n such that largest part k occurs at most floor(k/2) times.
+10
5
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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].
LINKS
Table of n, a(n) for n=1..46.
FORMULA
G.f.=sum(x^k*(1-x^(k(floor(k/2))))/product(1-x^j, j=1..k), k=1..infinity).
EXAMPLE
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].
MAPLE
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);
MATHEMATICA
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 *)
CROSSREFS
Cf. A118082, A118083.
Cf. A237758, A237757, A237799, A237800. - Clark Kimberling, Feb 15 2014
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 12 2006
STATUS
approved
A237758
Number of partitions of n such that 2*(least part) < number of parts.
+10
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Table of n, a(n) for n=1..47.
EXAMPLE
a(5) = 4 counts these partitions: 311, 221, 2111, 11111.
MATHEMATICA
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 *)
CROSSREFS
Cf. A118084, A237757, A237799, A237800.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 15 2014
STATUS
approved
A237799
Number of partitions of n such that 2*(least part) > number of parts.
+10
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Table of n, a(n) for n=1..55.
EXAMPLE
a(7) = 4 counts these partitions: 7, 52, 43, 322.
MATHEMATICA
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 *)
CROSSREFS
Cf. A237758, A118084, A237757, A237800.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 15 2014
STATUS
approved
A362049
Number of integer partitions of n such that (length) = 2*(median).
+10
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..57.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#]==2*Median[#]&]], {n, 30}]
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 10 2023
STATUS
approved
A362048
Number of integer partitions of n such that (length) <= 2*(median).
+10
2
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
LINKS
Table of n, a(n) for n=1..50.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#]<=2*Median[#]&]], {n, 30}]
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 10 2023
STATUS
approved
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