A116930 Sum of parts, counted without multiplicities, in all partitions of n into odd parts.

1, 1, 4, 5, 10, 14, 22, 31, 44, 61, 82, 111, 145, 191, 245, 316, 399, 506, 631, 788, 973, 1200, 1468, 1792, 2174, 2630, 3167, 3802, 4547, 5422, 6445, 7638, 9029, 10642, 12515, 14679, 17181, 20061, 23379, 27185, 31554, 36551, 42268, 48787, 56224, 64681, 74300

Offset: 1

Author

Emeric Deutsch, Feb 27 2006

Keywords

• nonn

Examples

			a(5) = 10 because the partitions of 5 into odd parts are [5], [3,1,1] and [1,1,1,1,1], with sum of the parts, counted without multiplicities 5 + (3+1) + 1 = 10.
a(5) = 10: There are three partitions of 5 into distinct parts, namely [5], [4,1], and [3,2]. We have phi(5) + phi(4) + phi(1) + phi(3) + phi(2) = 4 + 2 + 1 + 2 + 1 = 10. - _Peter Bala_, Dec 26 2013
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..7500

Crossrefs

Cf. A000010, A116929, A014153.

Programs

Maple

f:=x*(1+x^2)/(1-x^2)^2/product(1-x^(2*j-1),j=1..40): fser:=series(f,x=0,55): seq(coeff(fser,x^n),n=1..49);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(n>i*(i+1)/2, 0, b(n, i-1)+(p-> p+[0, p[1]
       *numtheory[phi](i)])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 15 2021

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1, 0},
     If[n > i (i + 1)/2, {0, 0}, b[n, i - 1] +
     With[{p = b[n - i, Min[n-i, i-1]]}, p + {0, p[[1]]*EulerPhi[i]}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 13 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A116929(n,k).

G.f.: x(1+x^2)/[(1-x^2)^2*product(1-x^(2*j-1),j=1..infinity)].

a(n) = Sum_{parts k in all partitions of n into distinct parts} phi(k), where phi(k) is the Euler totient function (see A000010). An example is given below. - Peter Bala, Dec 26 2013

a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/3)) * n^(1/4) / (2*Pi^2). - Vaclav Kotesovec, Jun 11 2025