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a(n) is the number of partitions of n with Durfee square of size <= 4.
2

%I #20 Jan 01 2020 15:20:43

%S 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,

%T 1002,1255,1575,1957,2434,3005,3708,4545,5568,6779,8245,9974,12046,

%U 14478,17372,20747,24732,29360,34782,41045,48337,56716,66410,77498,90247,104763,121366,140181,161590,185755

%N a(n) is the number of partitions of n with Durfee square of size <= 4.

%F a(n) = A000041(n), 0 <= n <= 24.

%F a(n) = A330641(n), 0 <= n <= 15.

%F a(n) = A330641(n) + A117486(n-16), n >= 16.

%F a(n) = n + A006918(n-3) + A117485(n) + A117486(n-16), n >= 16.

%F Conjectures from _Colin Barker_, Jan 01 2020: (Start)

%F G.f.: (1 - x - x^2 + 3*x^5 - x^7 - 2*x^8 - 2*x^9 + 3*x^10 + x^11 + x^12 - x^13 - 2*x^14 + x^15 + x^17 - x^19 + x^20) / ((1 - x)^8*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^2).

%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) - 4*a(n-5) + 4*a(n-6) + 4*a(n-7) + 2*a(n-8) - 10*a(n-10) + 2*a(n-12) + 4*a(n-13) + 4*a(n-14) - 4*a(n-15) - a(n-16) - 2*a(n-17) + a(n-18) + 2*a(n-19) - a(n-20) for n>20.

%F (End)

%Y Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A117486, A330640, A330641, A330643.

%K nonn

%O 0,3

%A _Omar E. Pol_, Dec 24 2019