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A186412
Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).
32
1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 2, 6, 3, 13, 5, 4, 38, 3, 7, 4, 14, 3, 9, 5, 49, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 69, 3, 7, 4, 14, 3, 9, 5, 27, 5, 4, 15, 7, 6, 87, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 39, 3, 9, 5, 19, 4, 12, 7, 6, 123
OFFSET
1,2
COMMENTS
Also triangle read by rows: T(j,k) = sum of all parts in the k-th region of the last section of the set of partitions of j. See Example section. For more information see A135010. - Omar E. Pol, Nov 26 2011
For the definition of "region" see A206437. - Omar E. Pol, Aug 19 2013
FORMULA
a(A000041(n)) = A046746(n).
EXAMPLE
Contribution from Omar E. Pol, Nov 26 2011 (Start):
Written as a triangle:
1;
3;
5;
2,9;
3,12;
2,6,3,20;
3,7,4,25;
2,6,3,13,5,4,38;
3,7,4,14,3,9,5,49;
2,6,3,13,5,4,23,4,10,6,5,69;
3,7,4,14,3,9,5,27,5,4,15,7,6,87;
2,6,3,13,5,4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123;
(End)
From Omar E. Pol, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
. _ _ _ _ _
. _ _ _ |_ _ _ _ _|
. _ _ _ _ |_ _ _| |_ _|
. _ _ |_ _ _ _| |_|
. _ _ _ |_ _| |_ _| |_|
. _ _ |_ _ _| |_| |_|
. _ |_ _| |_| |_| |_|
. |_| |_| |_| |_| |_|
.
. 1 3 5 2 9 3 12
.
(End)
MATHEMATICA
lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0, 2];
A186412 = {}; l = {};
For[j = 1, j <= 50, j++,
mx = Max@lex[j][[j]]; AppendTo[l, mx];
For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
];
A186412 (* Robert Price, Jul 25 2020 *)
CROSSREFS
Row sums of triangle A186114 and of triangle A193870.
Row j has length A187219(j).
Row sums give A138879.
Right border gives A046746, j >= 1.
Records give A046746, j >= 1.
Partial sums give A182244.
Sequence in context: A075626 A152649 A327263 * A322982 A275846 A273668
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 12 2011
STATUS
approved