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%I #94 Feb 04 2014 08:57:02
%S 0,1,2,2,3,3,4,3,4,4,5,4,5,5,6,4,5,5,6,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,
%T 7,7,8,7,8,8,8,9,6,7,7,8,7,8,8,9,8,8,9,9,9,10,6,7,7,8,8,8,9,8,9,9,9,
%U 10,8,9,9,10,9,10,10,10,11,7,8,8,9,8,9
%N Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j).
%C a(n) is also the column number in which is located the part of size 1 in the n-th zone of the tail of the last section of the set of partitions of k in colexicographic order, minus the column number in which is located the part of size 1 in the first row of the same tail, when k -> infinity (see example). For the definition of "section" see A135010.
%H Alois P. Heinz, <a href="/A207034/b207034.txt">Table of n, a(n) for n = 1..10143</a>
%F a(n) = t(n) - A194548(n), if n >= 2, where t(n) is the n-th element of the following sequence: triangle read by rows in which row n lists n repeated k times, where k = A187219(n).
%F a(n) = A000120(A194602(n-1)) = A000120(A228354(n)-1).
%F a(n) = i - A193173(i,n), i >= 1, 1<=n<=A000041(i).
%e Illustration of initial terms, n = 1..15. Consider the last 15 rows of the tail of the last section of the set of partitions in colexicographic order of any integer >= 8. The tail contains at least A000041(8-1) = 15 parts of size 1. a(n) is also the number of dots in the n-th row of the diagram.
%e ----------------------------------
%e n Tail a(n)
%e ----------------------------------
%e 15 1 . . . . . . 6
%e 14 1 . . . . . 5
%e 13 1 . . . . . 5
%e 12 1 . . . . 4
%e 11 1 . . . . . 5
%e 10 1 . . . . 4
%e 9 1 . . . . 4
%e 8 1 . . . 3
%e 7 1 . . . . 4
%e 6 1 . . . 3
%e 5 1 . . . 3
%e 4 1 . . 2
%e 3 1 . . 2
%e 2 1 . 1
%e 1 1 0
%e ----------------------------------
%e Written as a triangle:
%e 0;
%e 1;
%e 2;
%e 2,3;
%e 3,4;
%e 3,4,4,5;
%e 4,5,5,6;
%e 4,5,5,6,6,6,7;
%e 5,6,6,7,6,7,7,8;
%e 5,6,6,7,7,7,8,7,8,8,8,9;
%e 6,7,7,8,7,8,8,9,8,8,9,9,9,10;
%e 6,7,7,8,8,8,9,8,9,9,9,10,8,9,9,10,9,10,10,10,11;
%e ...
%e Consider a matrix [j X A000041(j)] in which the rows represent the partitions of j in colexicographic order (see A211992). Every part of every partition is located in a cell of the matrix. We can see that a(n) is the number of empty cells in row n for any integer j, if A000041(j) >= n. The number of empty cells in row n equals the sum of all parts minus the number of parts in the n-th partition of j.
%e Illustration of initial terms. The smallest part of every partition is located in the last column of the matrix.
%e ---------------------------------------------------------
%e . j: 1 2 3 4 5 6
%e n a(n)
%e ---------------------------------------------------------
%e 1 0 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 2 1 | . 2 . 2 1 . 2 1 1 . 2 1 1 1 . 2 1 1 1 1
%e 3 2 | . . 3 . . 3 1 . . 3 1 1 . . 3 1 1 1
%e 4 2 | . . 2 2 . . 2 2 1 . . 2 2 1 1
%e 5 3 | . . . 4 . . . 4 1 . . . 4 1 1
%e 6 3 | . . . 3 2 . . . 3 2 1
%e 7 4 | . . . . 5 . . . . 5 1
%e 8 3 | . . . 2 2 2
%e 9 4 | . . . . 4 2
%e 10 4 | . . . . 3 3
%e 11 5 | . . . . . 6
%e ...
%e Illustration of initial terms. In this case the largest part of every partition is located in the first column of the matrix.
%e ---------------------------------------------------------
%e . j: 1 2 3 4 5 6
%e n a(n)
%e ---------------------------------------------------------
%e 1 0 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 2 1 | 2 . 2 1 . 2 1 1 . 2 1 1 1 . 2 1 1 1 1 .
%e 3 2 | 3 . . 3 1 . . 3 1 1 . . 3 1 1 1 . .
%e 4 2 | 2 2 . . 2 2 1 . . 2 2 1 1 . .
%e 5 3 | 4 . . . 4 1 . . . 4 1 1 . . .
%e 6 3 | 3 2 . . . 3 2 1 . . .
%e 7 4 | 5 . . . . 5 1 . . . .
%e 8 3 | 2 2 2 . . .
%e 9 4 | 4 2 . . . .
%e 10 4 | 3 3 . . . .
%e 11 5 | 6 . . . . .
%e ...
%Y Row r has length A187219(r). Partial sums give A207038. Row sums give A207035. Right border gives A001477. Where records occur give A000041 without repetitions.
%Y Cf. A135010, A138121, A141285, A182703, A194548, A196087, A207031, A207032, A207035, A211992, A228716, A230440.
%K nonn,tabf
%O 1,3
%A _Omar E. Pol_, Feb 20 2012