A026923 -id:A026923 - cp's OEIS frontend

A309688 Sum of the odd parts appearing among the second largest parts of the partitions of n into 3 parts.

0, 0, 0, 1, 1, 1, 1, 4, 7, 10, 10, 15, 20, 25, 30, 42, 49, 56, 63, 79, 95, 111, 120, 140, 160, 180, 200, 233, 257, 281, 305, 344, 383, 422, 450, 495, 540, 585, 630, 694, 745, 796, 847, 919, 991, 1063, 1120, 1200, 1280, 1360, 1440, 1545, 1633, 1721, 1809

Offset: 0

Wesley Ivan Hurt, Aug 12 2019

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

Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,1,-4,6,-8,6,-4,1,2,-3,4,-3,2,-1).

Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309689, A309690, A309692, A309694.

Table[Sum[Sum[i * Mod[i, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{2, -3, 4, -3, 2, 1, -4, 6, -8, 6, -4, 1, 2, -3, 4, -3, 2, -1}, {0, 0, 0, 1, 1, 1, 1, 4, 7, 10, 10, 15, 20, 25, 30, 42, 49, 56}, 80]

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} i * (i mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: x^3*(1 + x + x^2 + x^3 + x^4)*(1 - 2*x + 3*x^2 - 4*x^3 + 6*x^4 - 4*x^5 + 3*x^6 - 2*x^7 + x^8) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 3*a(n-4) + 2*a(n-5) + a(n-6) - 4*a(n-7) + 6*a(n-8) - 8*a(n-9) + 6*a(n-10) - 4*a(n-11) + a(n-12) + 2*a(n-13) - 3*a(n-14) + 4*a(n-15) - 3*a(n-16) + 2*a(n-17) - a(n-18) for n>17.
(End)

A026920 Triangular array O by rows: O(n,k) = number of partitions of n into an odd number of parts, the greatest being k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 1, 3, 1, 1, 0, 1, 0, 2, 2, 3, 1, 1, 0, 1, 1, 2, 4, 3, 3, 1, 1, 0, 1, 0, 3, 3, 5, 3, 3, 1, 1, 0, 1, 1, 2, 6, 5, 6, 3, 3, 1, 1, 0, 1, 0, 3, 5, 8, 6, 6, 3, 3, 1, 1, 0, 1, 1, 3, 8, 8, 10, 7, 6, 3, 3, 1, 1, 0, 1, 0, 4, 7, 12, 10, 11, 7, 6, 3, 3, 1, 1, 0, 1

Offset: 1

Clark Kimberling

The reversed rows (see example) stabilize to A027187. [Joerg Arndt, May 12 2013]

			G.f. = (0)*q^0 +
(1) * q^1
(0* + 1*z^1) * q^2
(1* + 0*z^1 + 1*z^2) * q^3
(0* + 1*z^1 + 0*z^2 + 1*z^3) * q^4
(1* + 1*z^1 + 1*z^2 + 0*z^3 + 1*z^4) * q^5
(0* + 2*z^1 + 1*z^2 + 1*z^3 + 0*z^4 + 1*z^5) * q^6
(1* + 1*z^1 + 3*z^2 + 1*z^3 + 1*z^4 + 0*z^5 + 1*z^6) * q^7
... [_Joerg Arndt_, May 12 2013]
Triangle starts:
01: [1]
02: [0, 1]
03: [1, 0, 1]
04: [0, 1, 0, 1]
05: [1, 1, 1, 0, 1]
06: [0, 2, 1, 1, 0, 1]
07: [1, 1, 3, 1, 1, 0, 1]
08: [0, 2, 2, 3, 1, 1, 0, 1]
09: [1, 2, 4, 3, 3, 1, 1, 0, 1]
10: [0, 3, 3, 5, 3, 3, 1, 1, 0, 1]
11: [1, 2, 6, 5, 6, 3, 3, 1, 1, 0, 1]
12: [0, 3, 5, 8, 6, 6, 3, 3, 1, 1, 0, 1]
13: [1, 3, 8, 8, 10, 7, 6, 3, 3, 1, 1, 0, 1]
14: [0, 4, 7, 12, 10, 11, 7, 6, 3, 3, 1, 1, 0, 1]
15: [1, 3, 11, 13, 16, 12, 12, 7, 6, 3, 3, 1, 1, 0, 1]
16: [0, 4, 9, 18, 17, 18, 13, 12, 7, 6, 3, 3, 1, 1, 0, 1]
17: [1, 4, 13, 19, 25, 21, 20, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]
18: [0, 5, 12, 24, 27, 30, 23, 21, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]
19: [1, 4, 17, 26, 37, 34, 34, 25, 22, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]
... [_Joerg Arndt_, May 12 2013]

O(n, k) = E(n-k, 1)+E(n-k, 2)+...+E(n-k, m), where m=MIN{k, n-k}, n >= 2, E given by A026921.

Columns k=2..6: A026922, A026923, A026924, A026925, A026926.

N = 20;  q = 'q + O('q^N);
gf = sum(n=0,N, q^(2*n+1)/prod(k=1, 2*n+1, 1-'z*q^k) );
v = Vec(gf);
{ for(n=1, #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-1, j, p[j]);
    print(p);
); }
/* Joerg Arndt, May 12 2013 */

G.f.: sum(n>=0, q^(2*n+1)/prod(k=1..2*n+1, 1-z*q^k) ), setting z=1 gives g.f. for A027193. [Joerg Arndt, May 12 2013]

O(n,k) + A026921(n,k) = A008284(n,k). - R. J. Mathar, Aug 23 2019

cp's OEIS Frontend

A309688 Sum of the odd parts appearing among the second largest parts of the partitions of n into 3 parts.

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