A110618 -id:A110618 - cp's OEIS frontend

A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.

0, 0, 1, 2, 3, 5, 9, 11, 18, 26, 35, 49, 70, 89, 123, 164, 212, 278, 366, 460, 597, 762, 957, 1210, 1530, 1891, 2369, 2943, 3621, 4468, 5507, 6703, 8210, 10004, 12115, 14688, 17782, 21365, 25743, 30913, 36965, 44210, 52801, 62753, 74667, 88626, 104874, 124070

Offset: 0

Gus Wiseman, Jul 06 2023

nonn

We use the "rounding half to even" rule, see link.

			The a(0) = 0 through a(8) = 18 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)     (7)     (8)
             (21)  (22)  (32)   (33)    (43)    (44)
                   (31)  (41)   (42)    (52)    (53)
                         (221)  (51)    (61)    (62)
                         (311)  (222)   (322)   (71)
                                (321)   (331)   (332)
                                (411)   (421)   (422)
                                (2211)  (511)   (431)
                                (3111)  (2221)  (521)
                                        (3211)  (611)
                                        (4111)  (2222)
                                                (3221)
                                                (3311)
                                                (4211)
                                                (5111)
                                                (22211)
                                                (32111)
                                                (41111)

Wikipedia, Rounding.

Rounding-up gives A000065.
Rounding-down gives A110618, ranks A344291.
For median instead of mean we appear to have A238495.
The complement is counted by A363947, ranks A363948.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A067538 counts partitions with integer mean, ranks A316413.
A124943 counts partitions by low median, high A124944.
Cf. A002865, A098859, A241131, A327482, A363723, A363724, A363731, A363946.

Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]>1&]],{n,0,30}]

a(n) = A000041(n) - A363947(n).

A368503 Number of partitions of an n-set into blocks of size <= n/2.

Original entry on oeis.org

1, 0, 1, 1, 10, 26, 166, 652, 3795, 18755, 112124, 648649, 4163743, 27216840, 190168577, 1376119903, 10468226150, 82744297014, 681863474058, 5830425411936, 51720008131148, 474821737584174, 4506628734688128, 44150936144057758, 445956917001833090, 4638564968368158592

Offset: 0

Ilya Gutkovskiy, Dec 27 2023

nonn

Cf. A000110, A110618, A368502.

Table[n! SeriesCoefficient[Exp[Sum[x^j/j!, {j, 1, Floor[n/2]}]], {x, 0, n}], {n, 0, 25}]

a(n) = n! * [x^n] exp( Sum_{1 <= j <= n/2} x^j / j! ).

A110619 Triangle of number of partitions of n with no part more than n/k; also partitions of n into n/k or fewer parts.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 3, 1, 1, 7, 3, 1, 1, 1, 11, 7, 4, 1, 1, 1, 15, 8, 4, 1, 1, 1, 1, 22, 15, 5, 5, 1, 1, 1, 1, 30, 18, 12, 5, 1, 1, 1, 1, 1, 42, 30, 14, 6, 6, 1, 1, 1, 1, 1, 56, 37, 16, 6, 6, 1, 1, 1, 1, 1, 1, 77, 58, 34, 19, 7, 7, 1, 1, 1, 1, 1, 1, 101, 71, 39, 21, 7, 7, 1, 1, 1, 1, 1, 1, 1, 135, 105

Offset: 1

Henry Bottomley, Aug 01 2005

			Rows start: 1; 2,1; 3,1,1; 5,3,1,1; 7,3,1,1,1; 11,7,4,1,1,1; etc.
T(7,3)=4 since 7 can be partitioned as 1+1+1+1+1+1+1, 2+1+1+1+1+1, 2+2+1+1+1, or 2+2+2+1 and also as 7, 6+1, 5+2, or 4+3.

First column is A000041, second is A110618.

T(n, k)=A008284(n+floor[n/k], floor[n/k]). T(0, k)=1; T(n, k)=0 for 0A000041(n); T(n, 2)=A110618(n).

A363261 The partial sums of the prime indices of n include half the sum of all prime indices of n.

Original entry on oeis.org

4, 9, 12, 16, 25, 30, 40, 48, 49, 63, 64, 70, 81, 84, 108, 112, 121, 144, 154, 160, 165, 169, 192, 198, 220, 256, 264, 270, 273, 286, 289, 325, 351, 352, 360, 361, 364, 390, 442, 448, 468, 480, 520, 529, 561, 567, 576, 595, 624, 625, 640, 646, 675, 714, 729

Offset: 1

Gus Wiseman, Jun 06 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
   4: {1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  25: {3,3}
  30: {1,2,3}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
  63: {2,2,4}
  64: {1,1,1,1,1,1}
  70: {1,3,4}
  81: {2,2,2,2}
  84: {1,1,2,4}

Partitions of this type are counted by A322439.
For parts instead of partial sums we have A344415, counted by A035363.
A025065 counts palindromic partitions, ranked by A265640.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A301987 lists numbers whose sum of prime indices equals their product.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
Cf. A001414, A106529, A327473, A327476, A334201, A340387, A360005.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MemberQ[Accumulate[prix[#]],Total[prix[#]]/2]&]

A368484 Number of compositions (ordered partitions) of n into parts not greater than n/2.

Original entry on oeis.org

1, 0, 1, 1, 5, 8, 24, 44, 108, 208, 464, 912, 1936, 3840, 7936, 15808, 32192, 64256, 129792, 259328, 521472, 1042432, 2091008, 4180992, 8375296, 16748544, 33525760, 67047424, 134156288, 268304384, 536739840, 1073463296, 2147205120, 4294377472, 8589344768

Offset: 0

Ilya Gutkovskiy, Dec 26 2023

Index entries for sequences related to compositions
Index entries for linear recurrences with constant coefficients, signature (2,4,-8,-4,8).

Cf. A011782, A110618, A258259.

CoefficientList[Series[(1 - 2 x - 3 x^2 + 7 x^3 + 3 x^4 - 6 x^5)/((1 - 2 x) (1 - 2 x^2)^2), {x, 0, 34}], x]
Join[{1}, LinearRecurrence[{2, 4, -8, -4, 8}, {0, 1, 1, 5, 8}, 34]]

G.f.: (1 - 2*x - 3*x^2 + 7*x^3 + 3*x^4 - 6*x^5) / ((1 - 2*x) * (1 - 2*x^2)^2).

a(n) = [x^n] 1 / (1 - Sum_{1 <= j <= n/2} x^j).

A368501 Number of compositions (ordered partitions) of n into distinct parts not greater than n/2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 6, 0, 6, 6, 36, 30, 66, 60, 120, 234, 318, 432, 666, 894, 1272, 2226, 2772, 3960, 5496, 7524, 10068, 13776, 22488, 27756, 39162, 51264, 70398, 91386, 124152, 158574, 247554, 301656, 416748, 537690, 730854, 929196, 1248798, 1576014, 2078328, 2956110

Offset: 0

Ilya Gutkovskiy, Dec 27 2023

nonn

			a(6) = 6 because we have [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2] and [3,2,1].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A008289, A032020, A072575, A110618, A258259, A368484.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n, iquo(n, 2), 0):
seq(a(n), n=0..45);  # Alois P. Heinz, Dec 28 2023

Table[Sum[Count[IntegerPartitions[n, {k}], _?(And[UnsameQ @@ #, AllTrue[#, # <= n/2 &]] &)] k!, {k, 0, n}], {n, 0, 45}]

a(n) = Sum_{k=1..floor(n/2)} A072575(n,k) for n>=1. - Alois P. Heinz, Dec 31 2023

cp's OEIS Frontend

A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.

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A368503 Number of partitions of an n-set into blocks of size <= n/2.

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A110619 Triangle of number of partitions of n with no part more than n/k; also partitions of n into n/k or fewer parts.

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A363261 The partial sums of the prime indices of n include half the sum of all prime indices of n.

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A368484 Number of compositions (ordered partitions) of n into parts not greater than n/2.

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A368501 Number of compositions (ordered partitions) of n into distinct parts not greater than n/2.

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