A340571 Number of partitions of n into 4 parts with at least one even part.

0, 0, 0, 0, 0, 1, 1, 3, 3, 6, 6, 11, 10, 18, 17, 27, 25, 39, 36, 54, 49, 72, 66, 94, 85, 120, 109, 150, 135, 185, 167, 225, 202, 270, 243, 321, 287, 378, 339, 441, 394, 511, 457, 588, 524, 672, 600, 764, 680, 864, 770, 972, 864, 1089, 969, 1215, 1079, 1350, 1200, 1495, 1326

Offset

0,8

Links

Andrew Howroyd, Table of n, a(n) for n = 0..10000

Index entries for sequences related to partitions

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

Formula

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

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

From Andrew Howroyd, Nov 11 2025: (Start)

a(2*n+1) = A026810(2*n+1); a(2*n) = A026810(2*n) - A026810(n + 2).

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

a(n) = a(n-2) + a(n-4) - 2*a(n-10) + a(n-16) + a(n-18) - a(n-20). - Wesley Ivan Hurt, Nov 29 2025

Example

a(5) = 1; [2,1,1,1];
a(7) = 3; [4,1,1,1], [3,2,1,1], [2,2,2,1];
a(9) = 6; [6,1,1,1], [5,2,1,1], [4,3,1,1], [4,2,2,1], [3,3,2,1], [3,2,2,2].

Mathematica

Table[Sum[Sum[Sum[1 - Mod[k, 2] Mod[j, 2] Mod[i, 2] Mod[n - i - k - j, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

Crossrefs

Cf. A026810, A340589.

Sequence in context: A363241 A220153 A219627 * A131942 A240204 A200905

Adjacent sequences: A340568 A340569 A340570 * A340572 A340573 A340574

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 11 2021

Status

approved