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A263235
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Number of triangular number parts in all partitions of n.
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3
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0, 1, 2, 5, 8, 14, 24, 37, 56, 85, 124, 178, 254, 354, 489, 671, 907, 1217, 1624, 2144, 2815, 3675, 4764, 6142, 7885, 10062, 12788, 16183, 20391, 25590, 32013, 39883, 49536, 61326, 75688, 93129, 114296, 139856, 170718, 207857, 252476, 305938, 369946, 446314, 537379
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OFFSET
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0,3
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COMMENTS
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a(n) = Sum_{k=0..n} k*A263234(n,k).
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LINKS
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FORMULA
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G.f.: Sum_{i>0} x^h(i)/(1-x^h(i)) / Product_{i>0} 1-x^i, where h(i) = i*(i+1)/2.
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EXAMPLE
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a(4) = 8 because the partitions of 4 are [4], [3',1'], [2,2], [2,1',1'], and [1',1',1',1'], where the triangular number parts are marked.
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MAPLE
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h:= proc (i) options operator, arrow: (1/2)*i*(i+1) end proc: g := (sum(x^h(i)/(1-x^h(i)), i = 1..100))/(product(1-x^i, i = 1..100)): hser:= series(g, x = 0, 55): seq(coeff(hser, x, n), n = 0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, 0, b(n, i-1) +`if`(i>n, 0, (p-> p+
`if`(issqr(8*i+1), [0, p[1]], 0))(b(n-i, i)))))
end:
a:= n-> b(n$2)[2]:
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, p + If[IntegerQ@Sqrt[8*i+1], {0, p[[1]]}, 0]][b[n-i, i]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)
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CROSSREFS
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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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