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 1-4 of 4 results.

A026921 Triangular array E by rows: E(n,k) = number of partitions of n into an even number of parts, the greatest being k.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 2, 1, 2, 1, 1, 0, 1, 2, 3, 2, 2, 1, 1, 0, 0, 2, 3, 3, 2, 2, 1, 1, 0, 1, 2, 5, 4, 4, 2, 2, 1, 1, 0, 0, 3, 4, 6, 4, 4, 2, 2, 1, 1, 0, 1, 3, 7, 7, 7, 5, 4, 2, 2, 1, 1, 0, 0, 3, 6, 10, 8, 7, 5, 4, 2, 2, 1, 1, 0, 1, 3, 9, 11, 13, 9, 8, 5, 4, 2, 2, 1, 1, 0
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

The reversed rows (see example) stabilize to A027193. [Joerg Arndt, May 12 2013]

Examples

			G.f. = (1)*q^0 +
(0) * q^1 +
(1 + 0*z) * q^2 +
(0 + 1*z + 0*z^2) * q^3 +
(1 + 1*z + 1*z^2 + 0*z^3) * q^4 +
(0 + 1*z + 1*z^2 + 1*z^3 + 0*z^4) * q^5 +
(1 + 1*z + 2*z^2 + 1*z^3 + 1*z^4 + 0*z^5) * q^6 +
(0 + 2*z + 1*z^2 + 2*z^3 + 1*z^4 + 1*z^5 + 0*z^6) * q^7 +
... [_Joerg Arndt_, May 12 2013]
Triangle starts:
01: [0]
02: [1, 0]
03: [0, 1, 0]
04: [1, 1, 1, 0]
05: [0, 1, 1, 1, 0]
06: [1, 1, 2, 1, 1, 0]
07: [0, 2, 1, 2, 1, 1, 0]
08: [1, 2, 3, 2, 2, 1, 1, 0]
09: [0, 2, 3, 3, 2, 2, 1, 1, 0]
10: [1, 2, 5, 4, 4, 2, 2, 1, 1, 0]
11: [0, 3, 4, 6, 4, 4, 2, 2, 1, 1, 0]
12: [1, 3, 7, 7, 7, 5, 4, 2, 2, 1, 1, 0]
13: [0, 3, 6, 10, 8, 7, 5, 4, 2, 2, 1, 1, 0]
14: [1, 3, 9, 11, 13, 9, 8, 5, 4, 2, 2, 1, 1, 0]
15: [0, 4, 8, 14, 14, 14, 9, 8, 5, 4, 2, 2, 1, 1, 0]
16: [1, 4, 12, 16, 20, 17, 15, 10, 8, 5, 4, 2, 2, 1, 1, 0]
17: [0, 4, 11, 20, 22, 23, 18, 15, 10, 8, 5, 4, 2, 2, 1, 1, 0]
18: [1, 4, 15, 23, 30, 28, 26, 19, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
19: [0, 5, 13, 28, 33, 37, 31, 27, 19, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
20: [1, 5, 18, 31, 44, 44, 43, 34, 28, 20, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
... [_Joerg Arndt_, May 12 2013]

Crossrefs

E(n, k) = O(n-k, 1)+O(n-k, 2)+...+O(n-k, m), where m=MIN{k, n-k}, n >= 2, O given by A026920.
Columns k=3..6: A026927, A026928, A026929, A026930.

Programs

  • PARI
    N = 20;  q = 'q + O('q^N);
gf = sum(n=0,N, q^(2*n)/prod(k=1, 2*n, 1-'z*q^k) );
v = Vec(gf);
{ for(n=2, #v, /* print triangle starting with row 1: */
    p = Pol('c0 +'cn*'z^n + v[n],'z);
    p = polrecip(p);
    p = Vec(p);
    p[1] -= 'c0;
    p = vector(#p-2, j, p[j]);
    print(p);
); }
/* Joerg Arndt, May 12 2013 */

Formula

G.f. (including term a(0)=1): sum(n>=0, q^(2*n)/prod(k=1..2*n, 1-z*q^k) ), set z=1 to obtain g.f. for A027187. [Joerg Arndt, May 12 2013]
A026920(n,k) + E(n,k) = A008284(n,k). - R. J. Mathar, Aug 23 2019

A026924 Number of partitions of n into an odd number of parts, the greatest being 4; also, a(n+7) = number of partitions of n+3 into an even number of parts, each <=4.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 3, 3, 5, 5, 8, 8, 12, 13, 18, 19, 24, 26, 33, 35, 43, 46, 55, 59, 69, 74, 86, 91, 104, 111, 126, 134, 150, 159, 177, 187, 207, 219, 241, 254, 277, 292, 318, 334, 362, 380, 410, 430, 462, 484, 519, 542, 579, 605
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

4th column of A026920.

Programs

  • Maple
    A026924 := proc(n)
    local a,p1,p2,p3,p4 ;
    a := 0 ;
    for p1 from 0 to n do
        for p2 from 0 to (n-p1)/2 do
            for p3 from op(1+modp(n-p1-2*p2,4),[0,3,2,1]) to (n-p1-2*p2)/3 by 4 do
                p4 := (n-p1-2*p2-3*p3)/4 ;
                if type(p4,'integer') and p4 >=1 and type(p1+p2+p3+p4,'odd') then
                    a := a+1 ;
                end if:
            end do:
        end do:
    end do:
    a;
end proc: # R. J. Mathar, Aug 22 2019

Formula

a(n) + A026928(n) = A026810(n). - R. J. Mathar, Aug 22 2019

A309793 Number of odd parts appearing among the second largest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 13, 17, 20, 24, 27, 32, 36, 42, 47, 54, 60, 68, 75, 85, 93, 103, 112, 124, 135, 149, 161, 176, 189, 205, 220, 239, 256, 276, 294, 316, 336, 360, 382, 408, 432, 460, 486, 517, 545, 577, 607, 642, 675, 713, 748
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 17 2019

Keywords

Examples

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |      2           3           5           6           8        ...
--------------------------------------------------------------------------

Links

Crossrefs

Cf. A309795, A309797, A026928.

Programs

  • Mathematica
    LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 13}, 50]
  • PARI
    concat([0,0,0,0], Vec(x^4*(1 - x + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)) + O(x^50))) \\ Colin Barker, Oct 10 2019

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (i mod 2).
From Colin Barker, Aug 18 2019: (Start)
G.f.: x^4*(1 - x + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)).
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) - a(n-14) + 2*a(n-15) - a(n-16) for n>15.
(End) [Recurrence verified by Wesley Ivan Hurt, Aug 24 2019]

A309795 Number of even parts appearing among the second largest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 3, 3, 4, 5, 7, 9, 12, 14, 17, 19, 23, 27, 32, 36, 42, 47, 54, 60, 68, 75, 84, 92, 103, 113, 125, 135, 148, 160, 175, 189, 206, 221, 239, 255, 275, 294, 316, 336, 360, 382, 408, 432, 460, 486, 516, 544, 577, 608, 643, 675, 712, 747
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 17 2019

Keywords

Examples

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |      3           3           4           5           7        ...
--------------------------------------------------------------------------

Links

Crossrefs

Cf. A309793, A309797, A026928.

Programs

  • Mathematica
    LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, 1, 2, 3, 3, 4, 5, 7, 9, 12, 14}, 50]
  • PARI
    concat([0,0,0,0,0,0], Vec(x^6*(1 - x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)) + O(x^70))) \\ Colin Barker, Oct 10 2019

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} ((i-1) mod 2).
From Colin Barker, Aug 18 2019: (Start)
G.f.: x^6*(1 - x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)).
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) - a(n-14) + 2*a(n-15) - a(n-16) for n>15.
(End) [Recurrence verified by Wesley Ivan Hurt, Aug 25 2019]
Showing 1-4 of 4 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.