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Number of integer partitions of n with 2 distinct parts, none appearing more than twice.
2

%I #20 Feb 22 2024 09:09:24

%S 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,

%T 30,25,32,28,34,30,37,31,40,34,41,36,45,37,47,40,49,42,52,43,55,46,56,

%U 48,60,49,62,52,64,54,67,55,70,58,71,60,75,61,77,64,79,66

%N Number of integer partitions of n with 2 distinct parts, none appearing more than twice.

%C The Heinz numbers of these partitions appear to be given by A296205.

%H Colin Barker, <a href="/A307370/b307370.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1,2,1,0,-1,-1).

%F From _Colin Barker_, Apr 08 2019: (Start)

%F 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)).

%F 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)

%F 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

%e The a(3) = 1 through a(10) = 10 partitions:

%e (21) (31) (32) (42) (43) (53) (54) (64)

%e (211) (41) (51) (52) (62) (63) (73)

%e (221) (411) (61) (71) (72) (82)

%e (311) (2211) (322) (332) (81) (91)

%e (331) (422) (441) (433)

%e (511) (611) (522) (442)

%e (3311) (711) (622)

%e (811)

%e (3322)

%e (4411)

%t Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2&&Max@@Length/@Split[#]<=2&]],{n,0,30}]

%o (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

%Y Cf. A000041, A000726, A002133, A004709, A006918, A007774, A117485, A296205, A325168.

%Y Cf. A056594, A061347.

%K nonn,easy

%O 0,5

%A _Gus Wiseman_, Apr 05 2019