cp's OEIS Frontend

A330242 Sum of largest emergent parts of the partitions of n.

%I A330242 #26 Dec 07 2019 00:46:07
%S A330242 0,0,0,2,3,9,12,24,33,54,72,112,144,210,273,379,485,661,835,1112,1401,
%T A330242 1825,2284,2944,3652,4645,5745,7223,8879,11080,13541,16760,20406,
%U A330242 25062,30379,37102,44761,54351,65347,78919,94517,113645,135603,162331,193088,230182,272916,324195,383169,453571
%N A330242 Sum of largest emergent parts of the partitions of n.
%C A330242 In other words: a(n) is the sum of the largest parts of all partitions of n that contain emergent parts.
%C A330242 The partitions of n that contain emergent parts are the partitions that contain neither 1 nor n as a part. All parts of these partitions are emergent parts except the last part of every partition.
%C A330242 For the definition of emergent part see A182699.
%F A330242 a(n) = A138137(n) - n.
%F A330242 a(n) = A207031(n,1) - n.
%e A330242 For n = 9 the diagram of
%e A330242 the partitions of 9 that
%e A330242 do not contain 1 as a part
%e A330242 is as shown below:           Partitions
%e A330242 .
%e A330242     |_ _ _|   |   |   |      [3, 2, 2, 2]
%e A330242     |_ _ _ _ _|   |   |      [5, 2, 2]
%e A330242     |_ _ _ _|     |   |      [4, 3, 2]
%e A330242     |_ _ _ _ _ _ _|   |      [7, 2]
%e A330242     |_ _ _|     |     |      [3, 3, 3]
%e A330242     |_ _ _ _ _ _|     |      [6, 3]
%e A330242     |_ _ _ _ _|       |      [5, 4]
%e A330242     |_ _ _ _ _ _ _ _ _|      [9]
%e A330242 .
%e A330242 Note that the above diagram is also the "head" of the last section of the set of partitions of 9, where the "tail" is formed by A000041(9-1)= 22 1's.
%e A330242 The diagram of the
%e A330242 emergent parts is as
%e A330242 shown below:                 Emergent parts
%e A330242 .
%e A330242     |_ _ _|   |   |          [3, 2, 2]
%e A330242     |_ _ _ _ _|   |          [5, 2]
%e A330242     |_ _ _ _|     |          [4, 3]
%e A330242     |_ _ _ _ _ _ _|          [7]
%e A330242     |_ _ _|     |            [3, 3]
%e A330242     |_ _ _ _ _ _|            [6]
%e A330242     |_ _ _ _ _|              [5]
%e A330242 .
%e A330242 The sum of the largest emergent parts is 3 + 5 + 4 + 7 + 3 + 6 + 5 = 33, so a(9) = 33.
%Y A330242 Cf. A000041, A002865, A006128, A135010, A138135, A138137, A141285, A182699, A182703, A182709, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A207031, A299474, A299475.
%K A330242 nonn
%O A330242 1,4
%A A330242 _Omar E. Pol_, Dec 06 2019