A265248 Sum of the 2nd smallest parts of all the partitions of n (2nd smallest part is defined to be 0 when the partition does not have at least 2 distinct parts).

0, 0, 2, 5, 14, 22, 43, 63, 97, 140, 201, 266, 371, 492, 638, 837, 1079, 1377, 1748, 2207, 2756, 3471, 4287, 5317, 6537, 8081, 9840, 12069, 14643, 17837, 21543, 26113, 31385, 37877, 45318, 54433, 64944, 77682, 92341, 109995, 130373, 154769, 182866, 216350, 254905, 300648, 353259, 415392, 486843, 570867

Offset

1,3

Comments

a(n) = Sum_{k>=0} k*A265247(n,k).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

G.f.: G(x) = Sum_{i>=1} x^i/(1-x^i)*Sum_{j>=i+1} j*x^j/Product_{k>=j}(1-x^k).

Example

a(4) = 5 because in [4], [3,1], [2,2], [2,1,1], [1,1,1,1] the 2nd smallest parts are 0,3,0,2,0, respectively.

Maple

g := add(x^i*add(j*x^j/mul(1-x^k, k = j .. 100), j = i+1 .. 100)/(1-x^i), i = 1 .. 100): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 1 .. 50);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0],
      `if`(i>n, 0, add((p-> `if`(t=1, p+[0, i*p[1]], p))(
       b(n-i*j, i+1, min(t+1, 2))), j=1..n/i)+b(n, i+1, t)))
    end:
a:= n-> b(n, 1, 0)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 31 2015

Mathematica

Table[Total@ Flatten@ Map[Take[DeleteDuplicates@ #, {-2}] &, Select[IntegerPartitions@ n, Total@ Differences@ # != 0 && Length@ # >= 2 &]], {n, 50}] (* Michael De Vlieger, Dec 24 2015 *)
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0},
     If[i > n, {0, 0}, Sum[If[t == 1, # + {0, i*#[[1]]}, #]&[
     b[n - i*j, i+1, Min[t+1, 2]]], {j, 1, n/i}] + b[n, i+1, t]]];
a[n_] := b[n, 1, 0][[2]];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz *)

Crossrefs

Cf. A265247.

Sequence in context: A216669 A015633 A176191 * A131661 A321287 A076664

Adjacent sequences: A265245 A265246 A265247 * A265249 A265250 A265251

Keyword

nonn

Author

Emeric Deutsch, Dec 24 2015

Status

approved