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%I #7 Apr 18 2024 11:27:23
%S 1,6,16,31,52,76,107,143,184,233,289,354,427,512,606,716,835,972,1122,
%T 1292,1476,1685,1909,2161,2432,2734,3057,3417,3799,4222,4673,5168,
%U 5693,6270,6879,7545,8249,9014,9821,10698,11619,12616,13665,14795,15981,17259
%N Number of partitions of n into 6 parts such that every i-th smallest part (counted with multiplicity) is different from i.
%H Alois P. Heinz, <a href="/A244242/b244242.txt">Table of n, a(n) for n = 27..1000</a>
%F Conjectures from _Chai Wah Wu_, Apr 18 2024: (Start)
%F a(n) = a(n-1) + a(n-2) - a(n-5) - 2*a(n-7) + a(n-9) + a(n-10) + a(n-11) + a(n-12) - 2*a(n-14) - a(n-16) + a(n-19) + a(n-20) - a(n-21) for n > 57.
%F G.f.: x^27*(-x^30 + 2*x^25 + 2*x^24 + 2*x^23 + 4*x^22 + 2*x^21 + x^20 - 9*x^19 - 12*x^18 - 16*x^17 - 12*x^16 + x^15 + 13*x^14 + 24*x^13 + 25*x^12 + 20*x^11 + 3*x^10 - 11*x^9 - 23*x^8 - 22*x^7 - 15*x^6 - 6*x^5 + 5*x^4 + 9*x^3 + 9*x^2 + 5*x + 1)/((x - 1)^6*(x + 1)^3*(x^2 + 1)*(x^2 - x + 1)*(x^2 + x + 1)^2*(x^4 + x^3 + x^2 + x + 1)). (End)
%Y Column k=6 of A238406.
%K nonn,changed
%O 27,2
%A _Alois P. Heinz_, Jun 23 2014
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