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

Author

Emeric Deutsch, Dec 24 2015

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..5000 (terms 1..1000 from Alois P. Heinz)

Crossrefs

Cf. A265247.

Programs

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 *)

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).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*sqrt(3)*n). - Vaclav Kotesovec, Jun 12 2025