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

0,3

Comments

a(n) = Sum_{k=0..n} k*A263234(n,k).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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.

Example

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.

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 *)

Crossrefs

Cf. A263234, A309536.

Sequence in context: A304025 A264395 A139218 * A017988 A282444 A103077

Adjacent sequences: A263232 A263233 A263234 * A263236 A263237 A263238

Keyword

nonn

Author

Emeric Deutsch, Nov 12 2015

Status

approved