A330641 a(n) is the number of partitions of n with Durfee square of size <= 3.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 230, 295, 380, 480, 607, 758, 943, 1161, 1426, 1733, 2100, 2525, 3023, 3595, 4261, 5017, 5888, 6874, 7996, 9258, 10687, 12281, 14073, 16066, 18288, 20747, 23478, 26482, 29801, 33442, 37441, 41811, 46596, 51801, 57478, 63639, 70329, 77567

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Formula

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

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

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

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

From Colin Barker, Dec 31 2019: (Start)

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).

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. (End)

Prog

(PARI) seq(n)={Vec((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) + O(x*x^n))} \\ Andrew Howroyd, Dec 27 2024

Crossrefs

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

Sequence in context: A332746 A242698 A364793 * A242699 A194439 A046054

Adjacent sequences: A330638 A330639 A330640 * A330642 A330643 A330644

Keyword

nonn,easy

Author

Omar E. Pol, Dec 22 2019

Status

approved