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.

Showing 1-3 of 3 results.

A239871 Number of strict partitions of n having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 6, 1, 9, 2, 12, 3, 16, 6, 20, 10, 25, 17, 30, 26, 36, 40, 43, 57, 51, 81, 61, 110, 74, 148, 91, 193, 113, 250, 144, 316, 184, 397, 239, 491, 311, 603, 407, 732, 530, 885, 692, 1061, 895, 1268, 1155, 1508, 1478, 1790
Offset: 0

Views

Author

Clark Kimberling, Mar 29 2014

Keywords

Comments

Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of n having 1 more odd part than even, so that there is an ordering of parts for which the odd and even parts alternate and the first and last terms are odd. Then c(n) = a(n+1) for n >= 0.

Examples

			a(11) counts these 4 partitions:  812, 614, 632, 452.

Links

Crossrefs

Cf. A239241, A239872, A239873, A239832.
Column k=-1 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, 1):
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], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t =  Table[p[n], {n, 0, 20}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[n]], {n, 0, 70}]  (* A239871 *)
(* Peter J. C. Moses, Mar 10 2014 *)
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, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

Formula

a(n) = [x^n y^(-1)] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1). - Alois P. Heinz, Apr 03 2014

A239872 Number of strict partitions of 2n having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6, 10, 17, 26, 40, 57, 81, 110, 148, 193, 250, 316, 397, 491, 603, 732, 885, 1061, 1268, 1508, 1790, 2120, 2510, 2970, 3517, 4170, 4950, 5887, 7013, 8371, 10005, 11979, 14353, 17217, 20654, 24785, 29725, 35637, 42672, 51046, 60962
Offset: 0

Views

Author

Clark Kimberling, Mar 29 2014

Keywords

Comments

Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of 2n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even. This sequence is nondecreasing, unlike A239871, of which it is a bisection; the other bisection is A239873.

Examples

			a(9) counts these 3 partitions of 18:  [18], [8,3,4,1,2], [6,5,4,1,2].

Links

Crossrefs

Cf. A239241, A239871, A239873, A239832.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)-n>0, 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(2*n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 01 2014
  • Mathematica
    d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 20}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[2 n]], {n, 0, 40}]  (* A239872 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2 || Abs[t]-n > 0, 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[2*n, 2*n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

A269837 Irregular triangle read by rows: even terms of A094728(n+1) divided by 4.

Original entry on oeis.org

1, 2, 4, 3, 6, 4, 9, 8, 5, 12, 10, 6, 16, 15, 12, 7, 20, 18, 14, 8, 25, 24, 21, 16, 9, 30, 28, 24, 18, 10, 36, 35, 32, 27, 20, 11, 42, 40, 36, 30, 22, 12, 49, 48, 45, 40, 33, 24, 13, 56, 54, 50, 44, 36, 26, 14, 64, 63, 60, 55, 48, 39, 28, 15
Offset: 0

Views

Author

Paul Curtz, Mar 06 2016

Keywords

Comments

See A264798 and A261046 for the Hydrogen atom and the Janet periodic table.
a(n) odd terms are again A264798.
Decomposition by multiplication i.e. a(n) = b(n)*c(n) by irregular triangle:
1, 1 1,
2, 1 2,
4, 3, 2, 1, 2, 3,
6, 4, = 2, 1, * 3, 4,
9, 8, 5, 3, 2, 1, 3, 4, 5,
12, 10, 6, 3, 2, 1, 4, 5, 6,
16, 15, 12, 7, 4, 3, 2, 1, 4, 5, 6, 7,
etc. etc. etc.
b(n) is duplicated A004736(n) or mirror of A122197(n+1). c(n) = A138099(n+1).
Decomposition by subtraction, a(n) = d(n) - e(n):
1, 1 0,
2, 2, 0,
4, 3, 4, 3, 0, 0,
6, 4, = 6, 5, - 0, 1,
9, 8, 5, 9, 8, 7, 0, 0, 2,
12, 10, 6, 12, 11, 10, 0, 1, 4,
16, 15, 12, 7, 16, 15, 14, 13, 0, 0, 2, 6,
20, 18, 14, 8, 20, 19, 18, 17, 0, 1, 4, 9,
etc. etc. etc.
d(n) is the natural numbers A000027 inverted by lines. e(n) will be studied (see A239873).
Sum of a(n) by diagonals: 1, 5, 13, 27, 48, ... . The third differences have the period 2: repeat 2, 1. See A002717.

Crossrefs

Cf. A000027, A000034, A002717, A004736, A094728, A122197, A138099, A167430, A239873, A261046, A264798.

Programs

Showing 1-3 of 3 results.

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