A240009 -id:A240009 - cp's OEIS frontend

A240015 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 6.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 33, 41, 53, 66, 84, 104, 130, 162, 198, 246, 300, 369, 445, 548, 655, 801, 956, 1160, 1378, 1666, 1970, 2368, 2796, 3340, 3933, 4679, 5494, 6505, 7626, 8987, 10511, 12346, 14404, 16856, 19631, 22893, 26606, 30939

Offset: 6

Alois P. Heinz, Mar 30 2014

nonn

With offset 12 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -6.

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -6):
seq(a(n), n=6..80);

A240016 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 54, 66, 86, 105, 133, 164, 203, 250, 308, 376, 458, 560, 675, 821, 986, 1192, 1423, 1716, 2037, 2445, 2894, 3455, 4076, 4849, 5700, 6752, 7921, 9342, 10930, 12850, 14994, 17566, 20456, 23884, 27752, 32311

Offset: 7

Alois P. Heinz, Mar 30 2014

nonn

With offset 14 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -7.

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=7 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -7):
seq(a(n), n=7..80);

A240017 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 8.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 87, 105, 135, 165, 206, 252, 313, 380, 466, 567, 688, 833, 1006, 1212, 1454, 1748, 2083, 2495, 2963, 3532, 4177, 4965, 5848, 6924, 8134, 9593, 11236, 13212, 15429, 18082, 21070, 24613, 28611, 33332

Offset: 8

Alois P. Heinz, Mar 30 2014

nonn

With offset 16 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -8.

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -8):
seq(a(n), n=8..80);

A240018 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 9.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 136, 165, 208, 253, 316, 382, 471, 571, 696, 840, 1019, 1224, 1474, 1768, 2114, 2527, 3010, 3582, 4247, 5042, 5951, 7040, 8285, 9766, 11454, 13465, 15742, 18448, 21516, 25136, 29241, 34073

Offset: 9

Alois P. Heinz, Mar 30 2014

nonn

With offset 18 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -9.

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=9 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -9):
seq(a(n), n=9..80);

A240019 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 10.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 209, 253, 318, 383, 474, 573, 701, 844, 1027, 1231, 1487, 1780, 2134, 2547, 3041, 3614, 4294, 5092, 6022, 7117, 8389, 9882, 11607, 13638, 15963, 18702, 21834, 25504, 29694, 34600

Offset: 10

Alois P. Heinz, Mar 30 2014

nonn

With offset 20 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -10.

			a(17) = 4: [6,1,1,1,1,1,1,1,1,1,1,1], [5,2,1,1,1,1,1,1,1,1,1,1], [4,3,1,1,1,1,1,1,1,1,1,1], [3,3,2,1,1,1,1,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=10 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -10):
seq(a(n), n=10..80);

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

Original entry on oeis.org

0, 5, 6, 17, 18, 20, 24, 43, 45, 46, 53, 54, 58, 65, 66, 68, 72, 80, 96, 139, 141, 142, 149, 150, 154, 163, 165, 166, 169, 172, 177, 178, 180, 184, 197, 198, 202, 209, 210, 212, 216, 226, 232, 257, 258, 260, 264, 272, 288, 320, 343, 347, 349, 350, 363, 365

Offset: 1

Gus Wiseman, Jun 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   0: ()
   5: (2,1)
   6: (1,2)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  24: (1,4)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  53: (1,2,2,1)
  54: (1,2,1,2)
  58: (1,1,2,2)
  65: (6,1)
  66: (5,2)
  68: (4,3)
  72: (3,4)
  80: (2,5)
  96: (1,6)

A subset of A001969 (evil numbers), complement A000069.
These compositions are counted by A098123, without multiplicity A242821.
The version for partitions is A325698, counted by A045931.
For partitions without multiplicity we have A325700, counted by A241638.
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Count[stc[#],?EvenQ]==Count[stc[#],?OddQ]&]

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A240015 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 6.

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A240016 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 7.

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A240017 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 8.

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Maple

A240018 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 9.

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Maple

A240019 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 10.

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Maple

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

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Mathematica