A307370 Number of integer partitions of n with 2 distinct parts, none appearing more than twice.
0, 0, 0, 1, 2, 4, 4, 6, 7, 7, 10, 10, 11, 12, 15, 13, 17, 16, 19, 18, 22, 19, 25, 22, 26, 24, 30, 25, 32, 28, 34, 30, 37, 31, 40, 34, 41, 36, 45, 37, 47, 40, 49, 42, 52, 43, 55, 46, 56, 48, 60, 49, 62, 52, 64, 54, 67, 55, 70, 58, 71, 60, 75, 61, 77, 64, 79, 66
Offset: 0
Examples
The a(3) = 1 through a(10) = 10 partitions:
(21) (31) (32) (42) (43) (53) (54) (64)
(211) (41) (51) (52) (62) (63) (73)
(221) (411) (61) (71) (72) (82)
(311) (2211) (322) (332) (81) (91)
(331) (422) (441) (433)
(511) (611) (522) (442)
(3311) (711) (622)
(811)
(3322)
(4411)
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,1,0,-1,-1).
Crossrefs
Programs
-
Mathematica
Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2&&Max@@Length/@Split[#]<=2&]],{n,0,30}] -
PARI
concat([0,0,0], Vec(x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Apr 08 2019
Formula
From Colin Barker, Apr 08 2019: (Start)
G.f.: x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = -a(n-1) + a(n-3) + 2*a(n-4) + a(n-5) - a(n-7) - a(n-8) for n>8. (End)
a(n) = (27*n + 3*(n - 7)*(-1)^n - 53 - 6*A056594(n) + 8*A061347(n))/24 for n > 0. - Stefano Spezia, Feb 20 2024
Comments