A182244 Sum of all parts of the shell model of partitions of A135010 with n regions.

1, 4, 9, 11, 20, 23, 35, 37, 43, 46, 66, 69, 76, 80, 105, 107, 113, 116, 129, 134, 138, 176, 179, 186, 190, 204, 207, 216, 221, 270, 272, 278, 281, 294, 299, 303, 326, 330, 340, 346, 351, 420, 423, 430, 434, 448, 451, 460, 465, 492, 497, 501, 516, 523, 529, 616

Offset

1,2

Links

Table of n, a(n) for n=1..56.

Omar E. Pol, Illustration of the seven regions of 5

Formula

a(A000041(k)) = A066186(k), k >= 1.

Example

The first four regions of the shell model of partitions are [1],[2, 1],[3, 1, 1],[2], so a(4) = (1)+(2+1)+(3+1+1)+(2) = 11.
Written as a triangle begins:
1;
4;
9;
11,  20;
23,  35;
37,  43, 46, 66;
69,  76, 80,105;
107,113,116,129,134,138,176;
179,186,190,204,207,216,221,270;
272,278,281,294,299,303,326,330,340,346,351,420;
423,430,434,448,451,460,465,492,497,501,516,523,529,616;
...
From Omar E. Pol, Aug 08 2013: (Start)
Illustration of initial terms:
.                                                _ _ _ _ _
.                                      _ _ _    |_ _ _    |
.                            _ _ _ _  |_ _ _|_  |_ _ _|_  |
.                    _ _    |_ _    | |_ _    | |_ _    | |
.            _ _ _  |_ _|_  |_ _|_  | |_ _|_  | |_ _|_  | |
.      _ _  |_ _  | |_ _  | |_ _  | | |_ _  | | |_ _  | | |
.  _  |_  | |_  | | |_  | | |_  | | | |_  | | | |_  | | | |
. |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
.
.  1    4      9       11       20        23        35
.
.                                          _ _ _ _ _ _
.                             _ _ _       |_ _ _      |
.                _ _ _ _     |_ _ _|_     |_ _ _|_    |
.   _ _         |_ _    |    |_ _    |    |_ _    |   |
.  |_ _|_ _ _   |_ _|_ _|_   |_ _|_ _|_   |_ _|_ _|_  |
.  |_ _ _    |  |_ _ _    |  |_ _ _    |  |_ _ _    | |
.  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  | |
.  |_ _    | |  |_ _    | |  |_ _    | |  |_ _    | | |
.  |_ _|_  | |  |_ _|_  | |  |_ _|_  | |  |_ _|_  | | |
.  |_ _  | | |  |_ _  | | |  |_ _  | | |  |_ _  | | | |
.  |_  | | | |  |_  | | | |  |_  | | | |  |_  | | | | |
.  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|_|
.
.       37           43           46           66
(End)

Mathematica

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0, 2];
A186412 = {}; l = {};
For[j = 1, j <= 56, 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]];
  ];
Accumulate@A186412  (* Robert Price, Jul 25 2020 *)

Crossrefs

Partial sums of A186412. Row j has length A187219(j). Right border gives A066186.

Cf. A000041, A135010, A138121, A141285, A182244, A182181, A182276, A186114, A206437, A210990.

Sequence in context: A365892 A312845 A312846 * A312847 A141365 A179055

Adjacent sequences: A182241 A182242 A182243 * A182245 A182246 A182247

Keyword

nonn,tabf

Author

Omar E. Pol, Apr 23 2012

Status

approved