cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

A239880 Number of strict partitions of n having an ordering in which no parts of equal parity are juxtaposed and the first and last terms have the same parity.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 11, 14, 15, 19, 22, 26, 30, 35, 42, 47, 56, 62, 76, 83, 100, 108, 132, 142, 171, 184, 222, 239, 284, 306, 363, 394, 460, 500, 581, 636, 730, 802, 914, 1010, 1139, 1262, 1415, 1577, 1753, 1956, 2163, 2423, 2663
Offset: 0

Views

Author

Clark Kimberling, Mar 29 2014

Keywords

Comments

A strict partition is one in which every part has multiplicity 1.
a(n) = A240021(n,-1) + A240021(n,1). - Alois P. Heinz, Apr 02 2014

Examples

			a(12) counts these 7 partitions:  [12], [9,2,1], [3,8,1], [7,4,1], [7,2,3], [5,6,1], [5,4,3].

Links

Crossrefs

Cf. A239833, A239835, A239881.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
       abs(t)-n>1, 0, `if`(n=0, abs(t), b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014
  • Mathematica
    d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p[n_] := p[n] = Select[d[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] == 1 &]; t =  Table[p[n], {n, 0, 12}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[n]], {n, 0, 60}]  (* A239880 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2 || Abs[t]-n>1, 0, If[n==0, Abs[t], b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + (2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

A240138 Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 2.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 3, 2, 3, 4, 4, 7, 4, 11, 5, 16, 6, 23, 8, 31, 11, 41, 16, 53, 24, 67, 35, 83, 52, 102, 74, 124, 106, 149, 146, 179, 201, 214, 268, 256, 357, 307, 463, 370, 599, 447, 759, 545, 959, 667, 1192, 822, 1477, 1017, 1806, 1265, 2203, 1575
Offset: 4

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 6 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -2.

Examples

			a(16) = 4: [15,1], [13,3], [11,5], [9,7].
a(17) = 7: [11,3,2,1], [9,5,2,1], [9,4,3,1], [8,5,3,1], [7,6,3,1], [7,5,4,1], [7,5,3,2].

Links

Crossrefs

Column k=2 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -2):
seq(a(n), n=4..80);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + 2 Mod[i, 2] - 1]]]];
a[n_] := b[n, n, -2];
a /@ Range[4, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^2] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

A240139 Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 3.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 4, 1, 5, 2, 7, 4, 8, 7, 10, 12, 12, 18, 14, 27, 17, 38, 21, 53, 26, 71, 33, 94, 44, 121, 58, 155, 79, 194, 107, 241, 146, 296, 197, 361, 267, 436, 355, 525, 472, 628, 618, 750, 805, 894, 1035, 1064, 1324, 1267, 1673, 1511, 2103, 1804
Offset: 9

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 12 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -3.

Examples

			a(20) = 2: [9,5,3,2,1], [7,5,4,3,1].
a(21) = 7: [17,3,1], [15,5,1], [13,7,1], [13,5,3], [11,9,1], [11,7,3], [9,7,5].

Links

Crossrefs

Column k=3 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -3):
seq(a(n), n=9..80);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + 2 Mod[i, 2] - 1]]]];
a[n_] := b[n, n, -3];
a /@ Range[9, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^3] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

A240140 Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 4.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 6, 1, 9, 2, 11, 4, 15, 7, 18, 12, 23, 19, 27, 29, 34, 42, 40, 60, 49, 83, 59, 113, 73, 150, 89, 197, 112, 254, 141, 324, 180, 408, 231, 509, 298, 629, 386, 771, 500, 938, 648, 1135, 835, 1365, 1076, 1634, 1376, 1949, 1755, 2317
Offset: 16

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 20 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -4.

Examples

			a(28) = 9: [19,5,3,1], [17,7,3,1], [15,9,3,1], [15,7,5,1], [13,11,3,1], [13,9,5,1], [13,7,5,3], [11,9,7,1], [11,9,5,3].
a(29) = 2: [11,7,5,3,2,1], [9,7,5,4,3,1].

Links

Crossrefs

Column k=4 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -4):
seq(a(n), n=16..80);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + 2 Mod[i, 2] - 1]]]];
a[n_] := b[n, n, -4];
a /@ Range[16, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^4] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

A240141 Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 5.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 10, 1, 13, 2, 18, 4, 23, 7, 30, 12, 37, 19, 47, 30, 57, 44, 70, 64, 85, 90, 103, 125, 124, 169, 150, 227, 181, 298, 220, 388, 268, 498, 328, 634, 404, 797, 500, 996, 622, 1232, 775, 1515, 971, 1849, 1216, 2245, 1527, 2708
Offset: 25

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 30 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -5.

Examples

			a(39) = 13: [23,7,5,3,1], [21,9,5,3,1], [19,11,5,3,1], [19,9,7,3,1], [17,13,5,3,1], [17,11,7,3,1], [17,9,7,5,1], [15,13,7,3,1], [15,11,9,3,1], [15,11,7,5,1], [15,9,7,5,3], [13,11,9,5,1], [13,11,7,5,3].
a(40) = 2: [13,9,7,5,3,2,1], [11,9,7,5,4,3,1].

Links

Crossrefs

Column k=5 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -5):
seq(a(n), n=25..100);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n - i, i - 1, t + 2 Mod[i, 2] - 1]]]];
a[n_] := b[n, n, -5];
a /@ Range[25, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^5] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

A240142 Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 6.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 14, 1, 20, 2, 26, 4, 35, 7, 44, 12, 58, 19, 71, 30, 90, 45, 110, 66, 136, 94, 164, 132, 201, 181, 240, 246, 291, 328, 348, 433, 419, 564, 501, 728, 605, 929, 726, 1177, 878, 1477, 1061, 1841, 1288, 2278, 1565, 2801
Offset: 36

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 42 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -6.

Examples

			a(50) = 14: [25,9,7,5,3,1], [23,11,7,5,3,1], [21,13,7,5,3,1], [21,11,9,5,3,1], [19,15,7,5,3,1], [19,13,9,5,3,1], [19,11,9,7,3,1], [17,15,9,5,3,1], [17,13,11,5,3,1], [17,13,9,7,3,1], [17,11,9,7,5,1], [15,13,11,7,3,1], [15,13,9,7,5,1], [15,11,9,7,5,3].
a(51) = 1: [13,11,9,7,5,3,2,1].

Links

Crossrefs

Column k=6 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -6):
seq(a(n), n=36..120);

Formula

a(n) = [x^n y^6] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

A240143 Number of partitions of n into distinct parts, 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, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 21, 1, 28, 2, 38, 4, 49, 7, 65, 12, 82, 19, 105, 30, 131, 45, 164, 67, 201, 96, 248, 136, 301, 188, 366, 258, 441, 347, 531, 463, 635, 609, 761, 795, 907, 1025, 1082, 1313, 1289, 1665, 1537, 2099, 1831, 2624
Offset: 49

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 56 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -7.

Examples

			a(59) = 7: [23,11,9,7,5,3,1], [21,13,9,7,5,3,1], [19,15,9,7,5,3,1], [19,13,11,7,5,3,1], [17,15,11,7,5,3,1], [17,13,11,9,5,3,1], [15,13,11,9,7,3,1].
a(70) = 4: [19,13,11,9,7,5,3,2,1], [17,15,11,9,7,5,3,2,1], [17,13,11,9,7,5,4,3,1], [15,13,11,9,7,6,5,3,1].

Links

Crossrefs

Column k=7 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -7):
seq(a(n), n=49..120);

Formula

a(n) = [x^n y^7] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

A240144 Number of partitions of n into distinct parts, 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, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 29, 1, 40, 2, 52, 4, 70, 7, 89, 12, 116, 19, 146, 30, 186, 45, 230, 67, 288, 97, 352, 138, 434, 192, 526, 265, 640, 359, 769, 482, 928, 639, 1107, 840, 1325, 1092, 1574, 1410, 1874, 1803, 2218, 2291
Offset: 64

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 72 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -8.

Examples

			a(70) = 3: [21,13,11,9,7,5,3,1], [19,15,11,9,7,5,3,1], [17,15,13,9,7,5,3,1].
a(85) = 2: [19,15,13,11,9,7,5,3,2,1], [17,15,13,11,9,7,5,4,3,1].

Links

Crossrefs

Column k=8 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -8):
seq(a(n), n=64..130);

Formula

a(n) = [x^n y^8] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

A240145 Number of partitions of n into distinct parts, 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, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 41, 1, 54, 2, 73, 4, 94, 7, 123, 12, 157, 19, 201, 30, 252, 45, 318, 67, 393, 97, 488, 139, 598, 194, 732, 269, 888, 366, 1078, 494, 1296, 658, 1558, 870, 1862, 1137, 2222, 1477, 2639, 1900, 3133
Offset: 81

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 90 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -9.

Examples

			a(87) = 3: [23,15,13,11,9,7,5,3,1], [21,17,13,11,9,7,5,3,1], [19,17,15,11,9,7,5,3,1].
a(102) = 1: [19,17,15,13,11,9,7,5,3,2,1].

Links

Crossrefs

Column k=9 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -9):
seq(a(n), n=81..145);

Formula

a(n) = [x^n y^9] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

A240146 Number of partitions of n into distinct parts, 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, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 55, 1, 75, 2, 97, 4, 128, 7, 164, 12, 212, 19, 267, 30, 340, 45, 423, 67, 530, 97, 653, 139, 807, 195, 984, 271, 1204, 370, 1456, 501, 1763, 670, 2117, 889, 2543, 1167, 3032, 1522, 3618
Offset: 100

Views

Author

Alois P. Heinz, Apr 02 2014

Keywords

Comments

With offset 110 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -10.

Examples

			a(104) = 2: [23,17,15,13,11,9,7,5,3,1], [21,19,15,13,11,9,7,5,3,1].
a(125) = 2: [23,19,17,15,13,11,9,7,5,3,2,1], [21,19,17,15,13,11,9,7,5,4,3,1].

Links

Crossrefs

Column k=10 of A240021.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -10):
seq(a(n), n=100..160);

Formula

a(n) = [x^n y^10] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).
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