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

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1957, 2434, 3005, 3708, 4545, 5568, 6779, 8245, 9974, 12046, 14478, 17372, 20747, 24732, 29360, 34782, 41045, 48337, 56716, 66410, 77498, 90247, 104763, 121366, 140181, 161590, 185755

Offset: 0

Author

Omar E. Pol, Dec 24 2019

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1,-4,4,4,2,0,-10,0,2,4,4,-4,-1,-2,1,2,-1).

Crossrefs

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

Programs

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

Formula

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

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

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

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

From Colin Barker, Jan 01 2020: (Start)

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).

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.

(End)

G.f.: Sum_{k=0..4} x^(k^2)/(Product_{j=1..k} (1 - x^j))^2. - Andrew Howroyd, Dec 27 2024