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a(n) is the number of partitions of n with Durfee square of size <= 3.
3

%I #15 Dec 31 2019 16:39:43

%S 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,230,295,380,480,607,758,

%T 943,1161,1426,1733,2100,2525,3023,3595,4261,5017,5888,6874,7996,9258,

%U 10687,12281,14073,16066,18288,20747,23478,26482,29801,33442,37441,41811,46596,51801,57478,63639,70329,77567

%N a(n) is the number of partitions of n with Durfee square of size <= 3.

%F a(n) = A000041(n), 0 <= n <= 15.

%F a(n) = A330640(n), 0 <= n <= 8.

%F a(n) = A330640(n) + A117485(n), n >= 9.

%F a(n) = n + A006918(n-3) + A117485(n), n >= 9.

%F Conjectures from _Colin Barker_, Dec 31 2019: (Start)

%F G.f.: (1 - x - x^2 + 2*x^4 + x^5 - 2*x^6 - x^7 + x^8 + x^9 - x^11 + x^12) / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2).

%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>12.

%F (End)

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

%K nonn

%O 0,3

%A _Omar E. Pol_, Dec 22 2019