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A350842 Number of integer partitions of n with no difference -2.

%I A350842 #10 Jan 27 2022 20:46:41
%S A350842 1,1,2,3,4,6,9,12,16,24,30,40,54,69,89,118,146,187,239,297,372,468,
%T A350842 575,711,880,1075,1314,1610,1947,2359,2864,3438,4135,4973,5936,7090,
%U A350842 8466,10044,11922,14144,16698,19704,23249,27306,32071,37639,44019,51457,60113
%N A350842 Number of integer partitions of n with no difference -2.
%H A350842 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A350842 The a(1) = 1 through a(7) = 12 partitions:
%e A350842   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A350842        (11)  (21)   (22)    (32)     (33)      (43)
%e A350842              (111)  (211)   (41)     (51)      (52)
%e A350842                     (1111)  (221)    (222)     (61)
%e A350842                             (2111)   (321)     (322)
%e A350842                             (11111)  (411)     (511)
%e A350842                                      (2211)    (2221)
%e A350842                                      (21111)   (3211)
%e A350842                                      (111111)  (4111)
%e A350842                                                (22111)
%e A350842                                                (211111)
%e A350842                                                (1111111)
%t A350842 Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],-2]&]],{n,0,30}]
%Y A350842 Heinz number rankings are in parentheses below.
%Y A350842 The version for no difference 0 is A000009.
%Y A350842 The version for subsets of prescribed maximum is A005314.
%Y A350842 The version for all differences < -2 is A025157, non-strict A116932.
%Y A350842 The version for all differences > -2 is A034296, strict A001227.
%Y A350842 The opposite version is A072670.
%Y A350842 The version for no difference -1 is A116931 (A319630), strict A003114.
%Y A350842 The multiplicative version is A350837 (A350838), strict A350840.
%Y A350842 The strict case is A350844.
%Y A350842 The complement for quotients is counted by A350846 (A350845).
%Y A350842 A000041 = integer partitions.
%Y A350842 A027187 = partitions of even length.
%Y A350842 A027193 = partitions of odd length (A026424).
%Y A350842 A323092 = double-free partitions (A320340), strict A120641.
%Y A350842 A325534 = separable partitions (A335433).
%Y A350842 A325535 = inseparable partitions (A335448).
%Y A350842 A350839 = partitions with a gap and conjugate gap (A350841).
%Y A350842 Cf. A000070, A000929, A001511, A003242, A007359, A018819, A040039, A045690, A045691, A101417, A154402, A323093.
%K A350842 nonn
%O A350842 0,3
%A A350842 _Gus Wiseman_, Jan 20 2022