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%I #13 Oct 10 2024 13:40:49
%S 0,0,0,0,0,1,1,2,3,4,5,7,8,13,18,26,34,46,57,72,87,110,133,165,201,
%T 246,291,349,407,481,559,653,754,875,1003,1154,1309,1496,1690,1913,
%U 2152,2423,2707,3032,3373,3763,4169,4627,5109,5643,6204,6825,7473,8197
%N Sum of the odd parts appearing among the fourth largest parts in the partitions of n into 5 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_37">Index entries for linear recurrences with constant coefficients</a>, signature (3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -1).
%F 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)} k * (k mod 2).
%F a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 2*a(n-8) - 2*a(n-9) + 6*a(n-10) - 10*a(n-11) + 12*a(n-12) - 12*a(n-13) + 12*a(n-14) - 12*a(n-15) + 11*a(n-16) - 9*a(n-17) + 4*a(n-18) + 4*a(n-19) - 9*a(n-20) + 11*a(n-21) - 12*a(n-22) + 12*a(n-23) - 12*a(n-24) + 12*a(n-25) - 10*a(n-26) + 6*a(n-27) - 2*a(n-28) - 2*a(n-29) + 4*a(n-30) - 4*a(n-31) + 4*a(n-32) - 4*a(n-33) + 4*a(n-34) - 4*a(n-35) + 3*a(n-36) - a(n-37) for n > 36.
%e Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
%e 1+1+1+1+10
%e 1+1+1+2+9
%e 1+1+1+3+8
%e 1+1+1+4+7
%e 1+1+1+5+6
%e 1+1+1+1+9 1+1+2+2+8
%e 1+1+1+2+8 1+1+2+3+7
%e 1+1+1+3+7 1+1+2+4+6
%e 1+1+1+4+6 1+1+2+5+5
%e 1+1+1+5+5 1+1+3+3+6
%e 1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
%e 1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
%e 1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
%e 1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
%e 1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
%e 1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
%e 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
%e 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
%e 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
%e 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
%e 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
%e 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
%e 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
%e --------------------------------------------------------------------------
%e n | 10 11 12 13 14 ...
%e --------------------------------------------------------------------------
%e a(n) | 5 7 8 13 18 ...
%e --------------------------------------------------------------------------
%t LinearRecurrence[{3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -1}, {0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 13, 18, 26, 34, 46, 57, 72, 87, 110, 133, 165, 201, 246, 291, 349, 407, 481, 559, 653, 754, 875, 1003, 1154, 1309}, 50]
%t Table[Total[Select[IntegerPartitions[n,{5}][[;;,4]],OddQ]],{n,0,60}] (* _Harvey P. Dale_, Oct 10 2024 *)
%Y Cf. A309879, A309881, A309882.
%K nonn,easy
%O 0,8
%A _Wesley Ivan Hurt_, Aug 21 2019