A182276 Sum of all parts minus the total number of parts of the shell model of partitions with n regions.

0, 1, 3, 4, 8, 10, 15, 16, 20, 22, 31, 33, 38, 41, 51, 52, 56, 58, 67, 71, 74, 90, 92, 97, 100, 110, 112, 119, 123, 142, 143, 147, 149, 158, 162, 165, 181, 184, 192, 197, 201, 228, 230, 235, 238, 248, 250, 257, 261, 280, 284, 287, 299, 305, 310, 341

Offset: 1

Author

Omar E. Pol, Apr 23 2012

Keywords

• nonn

Comments

For the definition of "region of n" see A206437.

Examples

			Written has a triangle:
0,
1,
3,
4,    8;
10,  15;
16,  20, 22, 31;
33,  38, 41, 51;
52,  56, 58, 67, 71, 74, 90;
92,  97,100,110,112,119,123,142;
143,147,149,158,162,165,181,184,192,197,201,228;
230,235,238,248,250,257,261,280,284,287,299,305,310,341;
		

Crossrefs

Row j has length A187219(j). Right border gives A196087.

Cf. A000041, A066186, A006128, A135010, A138121, A182181, A182244, A186114, A206437, A207035.

Programs

Mathematica

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
reg = {}; 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[]]];
  t = Take[Reverse[First /@ lex[mx]], j - i];
  AppendTo[reg, Total@t - Length@t]
  ];
Accumulate@reg  (* Robert Price, Jul 25 2020 *)

Formula

a(n) = A182244(n) - A182181(n).

a(A000041(n)) = A196087(n).