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A196039
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Total sum of the smallest part of every partition of every shell of n.
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2
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0, 1, 4, 9, 18, 30, 50, 75, 113, 162, 231, 318, 441, 593, 798, 1058, 1399, 1824, 2379, 3066, 3948, 5042, 6422, 8124, 10264, 12884, 16138, 20120, 25027, 30994, 38312, 47168, 57955, 70974, 86733, 105676, 128516, 155850, 188644, 227783, 274541
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OFFSET
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0,3
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COMMENTS
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Total sum of parts of all regions of n that contain 1 as a part. - Omar E. Pol, Mar 11 2012
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LINKS
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FORMULA
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EXAMPLE
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For n = 5 the seven partitions of 5 are:
5
3 + 2
4 + 1
2 + 2 + 1
3 + 1 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
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The five shells of 5 (see A135010 and also A138121), written as a triangle, are:
1
2, 1
3, 1, 1
4, (2, 2), 1, 1, 1
5, (3, 2), 1, 1, 1, 1, 1
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The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:
1
2, 1
3, 1, 1
4, 2, 1, 1, 1
5, 2, 1, 1, 1, 1, 1
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thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.
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MAPLE
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b:= proc(n, i) option remember;
`if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i)))
end:
a:= proc(n) option remember;
b(n, n) +`if`(n=0, 0, a(n-1))
end:
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i-1] + If[n < i, 0, b[n-i, i]]]; Accumulate[Table[b[n, n], {n, 0, 50}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
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CROSSREFS
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Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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