A330643 a(n) is the number of partitions of n with Durfee square of size <= 5.

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, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17976, 21635, 26010, 31175, 37318, 44547, 53109, 63153, 74996, 88850, 105113, 124078, 146256, 172032, 202056, 236844

Offset: 0

Author

Omar E. Pol, Dec 24 2019

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2, 1, -2, -1, -2, 0, 2, 6, 2, -3, -6, -5, -2, 3, 12, 3, -2, -5, -6, -3, 2, 6, 2, 0, -2, -1, -2, 1, 2, -1).

Crossrefs

Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A117486, A117487, A330640, A330641, A330642.

Programs

PARI

seq(n) = Vec(sum(k=0, 5, 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 <= 35.

a(n) = A330642(n), 0 <= n <= 24.

a(n) = A330642(n) + A117487(n-24), n >= 25.

a(n) = n + A006918(n-3) + A117485(n) + A117486(n-16) + A117487(n-24), n >= 25.

G.f.: Sum_{k=0..5} x^(k^2)/(Product_{j=1..k} (1 - x^j))^2. - Andrew Howroyd, Dec 27 2024