cp's OEIS Frontend

A348552 Number of integer partitions of n with the same alternating product as alternating sum.

%I A348552 #5 Nov 03 2021 09:08:29
%S A348552 0,1,1,2,2,4,5,7,8,12,14,19,23,31,36,46,55,69,83,100,122,144,175,203,
%T A348552 249,284,348,393,484,536,661,725,898,975,1208,1297,1614,1715,2136,
%U A348552 2251,2812,2939,3674,3813,4779,4920,6172,6315,7943,8070,10156,10263,12944
%N A348552 Number of integer partitions of n with the same alternating product as alternating sum.
%C A348552 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. In the case of a partition, this equals the number of odd parts in the conjugate partition.
%C A348552 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%e A348552 The a(1) = 1 through a(9) = 12 partitions:
%e A348552   1   2   3     4     5       6       7         8         9
%e A348552           111   211   221     42      322       332       333
%e A348552                       311     222     331       422       441
%e A348552                       11111   411     511       611       522
%e A348552                               21111   22111     4211      711
%e A348552                                       31111     22211     22221
%e A348552                                       1111111   41111     32211
%e A348552                                                 2111111   33111
%e A348552                                                           51111
%e A348552                                                           2211111
%e A348552                                                           3111111
%e A348552                                                           111111111
%e A348552 For example, we have 3 - 2 + 2 - 1 + 1 = 3 / 2 * 2 / 1 * 1 = 3, so the partition (3,2,2,1,1) is counted under a(9).
%t A348552 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A348552 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A348552 Table[Length[Select[IntegerPartitions[n],altprod[#]==ats[#]&]],{n,0,30}]
%Y A348552 The version for reverse-alternating sum (or product, or both) is A025065.
%Y A348552 Dominated by A347446.
%Y A348552 A000041 counts partitions with alternating sum 0.
%Y A348552 A027187 counts partitions of even length.
%Y A348552 A027193 counts partitions of odd length, ranked by A026424.
%Y A348552 A097805 counts compositions by sum and alternating sum.
%Y A348552 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A348552 A119620 counts partitions with alternating product 1, ranked by A028982.
%Y A348552 A124754 gives alternating sums of standard compositions.
%Y A348552 A277103 counts partitions with the same alternating sum as their conjugate.
%Y A348552 A345927 gives alternating sums of binary expansions.
%Y A348552 Cf. A000070, A000097, A001700, A025047, A236913, A325534, A325535, A339846, A344607, A345196, A347443, A347448.
%K A348552 nonn
%O A348552 0,4
%A A348552 _Gus Wiseman_, Oct 30 2021