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

0,5

Comments

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

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).

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

Example

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)

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A056594, A061347.

Sequence in context: A330812 A368200 A316945 * A352923 A014246 A026413

Adjacent sequences: A307367 A307368 A307369 * A307371 A307372 A307373

Keyword

nonn,easy

Author

Gus Wiseman, Apr 05 2019

Status

approved