A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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

Author

Omar E. Pol, Aug 12 2011

Keywords

• nonn
• tabf

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

Examples

			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)
		

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.

Cf. A000041, A002865, A066186, A135010, A138121, A138879, A194436, A194437, A194438, A194439, A194446, A194447.

Programs

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 *)

Formula

a(A000041(n)) = A046746(n).