A309831 Number of even parts appearing among the smallest parts of the partitions of n into 5 parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 24, 28, 36, 42, 52, 60, 73, 83, 99, 112, 132, 148, 172, 192, 221, 245, 279, 308, 348, 382, 429, 469, 523, 570, 632, 686, 757, 819, 899, 970, 1061, 1141, 1243, 1334, 1448, 1550, 1677, 1791, 1932

Offset

0,13

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,0,-1,-1,0,0,2,0,0,-1,-1,1).

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} ((l-1) mod 2).

From Colin Barker, Aug 19 2019: (Start)

G.f.: x^10 / ((1 - x)^5*(1 + x)^3*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).

a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-11) - a(n-12) + 2*a(n-15) - a(n-18) - a(n-19) + a(n-20) for n>19.

(End) [Recurrence verified by Wesley Ivan Hurt, Aug 24 2019]

Example

Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      1           1           2           3           5        ...
--------------------------------------------------------------------------

Mathematica

LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, 0, -1, -1, 0, 0, 2, 0,
  0, -1, -1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9,
  11, 15, 18}, 50]

Prog

(PARI) concat([0, 0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(x^10 / ((1 - x)^5*(1 + x)^3*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^80))) \\ Colin Barker, Oct 10 2019

Crossrefs

Cf. A309787, A309830, A309834.

Sequence in context: A008772 A351003 A008771 * A309830 A008770 A347548

Adjacent sequences: A309828 A309829 A309830 * A309832 A309833 A309834

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 19 2019

Status

approved