cp's OEIS Frontend

A330642 a(n) is the number of partitions of n with Durfee square of size <= 4.

%I A330642 #28 Mar 07 2025 12:58:50
%S A330642 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,
%T A330642 1002,1255,1575,1957,2434,3005,3708,4545,5568,6779,8245,9974,12046,
%U A330642 14478,17372,20747,24732,29360,34782,41045,48337,56716,66410,77498,90247,104763,121366,140181,161590,185755
%N A330642 a(n) is the number of partitions of n with Durfee square of size <= 4.
%H A330642 Andrew Howroyd, <a href="/A330642/b330642.txt">Table of n, a(n) for n = 0..10000</a>
%H A330642 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1,-4,4,4,2,0,-10,0,2,4,4,-4,-1,-2,1,2,-1).
%F A330642 a(n) = A000041(n), 0 <= n <= 24.
%F A330642 a(n) = A330641(n), 0 <= n <= 15.
%F A330642 a(n) = A330641(n) + A117486(n-16), n >= 16.
%F A330642 a(n) = n + A006918(n-3) + A117485(n) + A117486(n-16), n >= 16.
%F A330642 From _Colin Barker_, Jan 01 2020: (Start)
%F A330642 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 A330642 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 A330642 (End)
%F A330642 G.f.: Sum_{k=0..4} x^(k^2)/(Product_{j=1..k} (1 - x^j))^2. - _Andrew Howroyd_, Dec 27 2024
%o A330642 (PARI) seq(n) = Vec(sum(k=0, 4, x^(k^2)/prod(j=1, k, 1 - x^j)^2) + O(x*x^n)) \\ _Andrew Howroyd_, Dec 27 2024
%Y A330642 Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A117486, A330640, A330641, A330643.
%K A330642 nonn
%O A330642 0,3
%A A330642 _Omar E. Pol_, Dec 24 2019