A103628 Total sum of parts of multiplicity 1 in all partitions of n.

0, 1, 2, 6, 10, 21, 33, 59, 89, 145, 212, 325, 463, 680, 948, 1348, 1845, 2558, 3446, 4681, 6219, 8306, 10901, 14352, 18632, 24230, 31151, 40077, 51074, 65088, 82290, 103986, 130517, 163679, 204078, 254174, 314975, 389839, 480369, 591133, 724600, 886965

Offset: 0

Author

Vladeta Jovovic, Mar 25 2005

Keywords

• easy
• nonn

Comments

Total number of parts of multiplicity 1 in all partitions of n is A024786(n+1).

Equals A000041 convolved with A026741. - Gary W. Adamson, Jun 11 2009

Examples

			Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and a(4) = 0 + 2 + 0 + (1+3) + 4 = 10.
		

Links

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

Crossrefs

Cf. A026741. - Gary W. Adamson, Jun 11 2009

Column k=1 of A222730. - Alois P. Heinz, Mar 03 2013

Programs

Maple

gf:=x*(1+x+x^2)/(1-x^2)^2/product((1-x^k), k=1..500): s:=series(gf, x, 100): for n from 0 to 60 do printf(`%d,`,coeff(s, x, n)) od: # James Sellers, Apr 22 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], add((l->`if`(j=1, [l[1],
       l[2]+l[1]*i], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2013

Mathematica

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[3]]; a[0] = 0; Table[a[n], {n, 0, 50}]  (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

Formula

G.f.: x*(1+x+x^2)/(1-x^2)^2 /Product_{k>0}(1-x^k).

a(n) = A066186(n) - A194544(n). - Omar E. Pol, Nov 20 2011

a(n) = 3*A014153(n)/4 - 3*A000070(n)/4 - A270143(n+1)/4 + A087787(n)/4. - Vaclav Kotesovec, Nov 05 2016

a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2) * (1 - Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Nov 05 2016

Extensions

More terms from James Sellers, Apr 22 2005