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 11-13 of 13 results.

A282249 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k < j_k, j_k > j_{k+1} and all factors distinct.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 4, 4, 6, 5, 6, 8, 8, 9, 11, 10, 14, 15, 14, 14, 21, 18, 21, 25, 25, 30, 34, 33, 42, 45, 41, 55, 62, 58, 66, 79, 76, 94, 95, 97, 115, 131, 120, 148, 153, 159, 175, 203, 189, 226, 232, 243, 268, 299, 271, 340, 349, 363, 389
Offset: 0

Views

Author

Alois P. Heinz, Feb 09 2017

Keywords

Comments

Or number of partitions of n where part i has multiplicity < i and all multiplicities are distinct and different from all parts.

Examples

			a(0) = 1: the empty sum.
a(6) = 2: 1*6 = 2*3.
a(8) = 2: 1*8 = 2*4.
a(10) = 3: 1*10 = 2*5 = 1*4+2*3.
a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3.
a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4.
a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4.
a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2.
a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5.
a(25) = 14: 1*25 = 1*19+2*3 = 1*17+2*4 = 1*15+2*5 = 1*13+2*6 = 1*13+3*4 = 2*11+1*3 = 1*11+2*7 = 2*10+1*5 = 1*10+3*5 = 2*9+1*7 = 1*9+2*8 = 3*7+1*4 = 1*7+3*6.

Links

Crossrefs

Cf. A000009, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282379.

Programs

  • Maple
    h:= proc(n) option remember;
      (((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12
    end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
                b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
     `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
      min(n-i*j, i-1), s union {j})), j=1..min(i-1, n/i)))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);
  • Mathematica
    h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12;
g[n_, i_, s_] := If[n==0, 1, If[n>h[i], 0, b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j, Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i - 1, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)

A282379 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k <= j_k, j_k > j_{k+1} and all factors distinct with the exception that i_k = j_k is allowed.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 5, 9, 9, 8, 11, 15, 13, 17, 17, 19, 24, 29, 23, 33, 37, 39, 40, 53, 48, 62, 63, 71, 77, 94, 81, 110, 116, 122, 123, 156, 152, 185, 180, 200, 213, 259, 236, 287, 298, 325, 333, 404, 386, 450, 457, 506, 531, 615, 579, 679, 721
Offset: 0

Views

Author

Alois P. Heinz, Feb 13 2017

Keywords

Examples

			a(4) = 2: 1*4 = 2*2.
a(5) = 2: 1*5 = 2*2+1*1.
a(6) = 2: 1*6 = 2*3.
a(7) = 3: 1*7 = 2*3+1*1 = 1*3+2*2.
a(8) = 3: 1*8 = 2*4 = 1*4+2*2.
a(9) = 4: 1*9 = 1*5+2*2 = 2*4+1*1 = 3*3.
a(10) = 5: 1*10 = 1*6+2*2 = 2*5 = 1*4+2*3 = 3*3+1*1.
a(11) = 6: 1*11 = 1*7+2*2 = 2*5+1*1 = 1*5+2*3 = 2*4+1*3 = 3*3+1*2.
a(12) = 5: 1*12 = 1*8+2*2 = 2*6 = 1*6+2*3 = 3*4.

Links

Crossrefs

Cf. A000009, A000330, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282249.

Programs

  • Maple
    h:= proc(n) option remember;
      n*(n+1)*(2*n+1)/6
    end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
                b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
     `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
      min(n-i*j, i-1), s union {j})), j=1..min(i, n/i)))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);
  • Mathematica
    h[n_] := h[n] = n(n+1)(2n+1)/6;
g[n_, i_, s_ ] := If[n == 0, 1, If[n > h[i], 0,
     b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] +
     If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j,
     Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

A321375 Expansion of Product_{1 < i <= j <= k} 1/(1 - x^(i*j*k)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 5, 0, 1, 1, 4, 0, 2, 0, 9, 0, 2, 1, 12, 0, 3, 1, 16, 0, 7, 2, 20, 2, 6, 2, 36, 0, 13, 5, 37, 2, 21, 4, 60, 3, 23, 9, 80, 4, 35, 14, 106, 5, 58, 16, 137, 12, 66, 22, 210, 10, 100, 40, 238, 22, 147
Offset: 0

Views

Author

Seiichi Manyama, Nov 08 2018

Keywords

Links

Crossrefs

Cf. A122179, A182270, A321360.

Formula

Euler transform of A122179.
G.f.: Product_{k>0} 1/(1 - x^k)^A122179(k).
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.