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A263235 Number of triangular number parts in all partitions of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)