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

1,3

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A000010, A116929, A014153.

Sequence in context: A366864 A322610 A322468 * A073119 A002257 A101528

Adjacent sequences: A116927 A116928 A116929 * A116931 A116932 A116933

Keyword

nonn

Author

Emeric Deutsch, Feb 27 2006

Status

approved