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%I #27 Jul 25 2020 12:16:15
%S 1,4,9,11,20,23,35,37,43,46,66,69,76,80,105,107,113,116,129,134,138,
%T 176,179,186,190,204,207,216,221,270,272,278,281,294,299,303,326,330,
%U 340,346,351,420,423,430,434,448,451,460,465,492,497,501,516,523,529,616
%N Sum of all parts of the shell model of partitions of A135010 with n regions.
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%F a(A000041(k)) = A066186(k), k >= 1.
%e 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.
%e Written as a triangle begins:
%e 1;
%e 4;
%e 9;
%e 11, 20;
%e 23, 35;
%e 37, 43, 46, 66;
%e 69, 76, 80,105;
%e 107,113,116,129,134,138,176;
%e 179,186,190,204,207,216,221,270;
%e 272,278,281,294,299,303,326,330,340,346,351,420;
%e 423,430,434,448,451,460,465,492,497,501,516,523,529,616;
%e ...
%e From _Omar E. Pol_, Aug 08 2013: (Start)
%e Illustration of initial terms:
%e . _ _ _ _ _
%e . _ _ _ |_ _ _ |
%e . _ _ _ _ |_ _ _|_ |_ _ _|_ |
%e . _ _ |_ _ | |_ _ | |_ _ | |
%e . _ _ _ |_ _|_ |_ _|_ | |_ _|_ | |_ _|_ | |
%e . _ _ |_ _ | |_ _ | |_ _ | | |_ _ | | |_ _ | | |
%e . _ |_ | |_ | | |_ | | |_ | | | |_ | | | |_ | | | |
%e . |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
%e .
%e . 1 4 9 11 20 23 35
%e .
%e . _ _ _ _ _ _
%e . _ _ _ |_ _ _ |
%e . _ _ _ _ |_ _ _|_ |_ _ _|_ |
%e . _ _ |_ _ | |_ _ | |_ _ | |
%e . |_ _|_ _ _ |_ _|_ _|_ |_ _|_ _|_ |_ _|_ _|_ |
%e . |_ _ _ | |_ _ _ | |_ _ _ | |_ _ _ | |
%e . |_ _ _|_ | |_ _ _|_ | |_ _ _|_ | |_ _ _|_ | |
%e . |_ _ | | |_ _ | | |_ _ | | |_ _ | | |
%e . |_ _|_ | | |_ _|_ | | |_ _|_ | | |_ _|_ | | |
%e . |_ _ | | | |_ _ | | | |_ _ | | | |_ _ | | | |
%e . |_ | | | | |_ | | | | |_ | | | | |_ | | | | |
%e . |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
%e .
%e . 37 43 46 66
%e (End)
%t lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
%t A186412 = {}; l = {};
%t For[j = 1, j <= 56, j++,
%t mx = Max@lex[j][[j]]; AppendTo[l, mx];
%t For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
%t AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
%t ];
%t Accumulate@A186412 (* _Robert Price_, Jul 25 2020 *)
%Y Partial sums of A186412. Row j has length A187219(j). Right border gives A066186.
%Y Cf. A000041, A135010, A138121, A141285, A182244, A182181, A182276, A186114, A206437, A210990.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Apr 23 2012
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