A263235 Number of triangular number parts in all partitions of n.
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
Offset: 0
Keywords
Examples
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.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Programs
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Maple
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]: seq(a(n), n=0..60); # Alois P. Heinz, Nov 13 2015 -
Mathematica
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 *)
Formula
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.
Comments