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

%I A116930 #20 Jun 11 2025 16:02:24
%S A116930 1,1,4,5,10,14,22,31,44,61,82,111,145,191,245,316,399,506,631,788,973,
%T A116930 1200,1468,1792,2174,2630,3167,3802,4547,5422,6445,7638,9029,10642,
%U A116930 12515,14679,17181,20061,23379,27185,31554,36551,42268,48787,56224,64681,74300
%N A116930 Sum of parts, counted without multiplicities, in all partitions of n into odd parts.
%H A116930 Alois P. Heinz, <a href="/A116930/b116930.txt">Table of n, a(n) for n = 1..7500</a>
%F A116930 a(n) = Sum_{k=1..n} k * A116929(n,k).
%F A116930 G.f.: x(1+x^2)/[(1-x^2)^2*product(1-x^(2*j-1),j=1..infinity)].
%F A116930 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
%F A116930 a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/3)) * n^(1/4) / (2*Pi^2). - _Vaclav Kotesovec_, Jun 11 2025
%e A116930 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.
%e A116930 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
%p A116930 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);
%p A116930 # second Maple program:
%p A116930 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A116930       `if`(n>i*(i+1)/2, 0, b(n, i-1)+(p-> p+[0, p[1]
%p A116930        *numtheory[phi](i)])(b(n-i, min(n-i, i-1)))))
%p A116930     end:
%p A116930 a:= n-> b(n$2)[2]:
%p A116930 seq(a(n), n=1..50);  # _Alois P. Heinz_, Aug 15 2021
%t A116930 b[n_, i_] := b[n, i] = If[n == 0, {1, 0},
%t A116930      If[n > i (i + 1)/2, {0, 0}, b[n, i - 1] +
%t A116930      With[{p = b[n - i, Min[n-i, i-1]]}, p + {0, p[[1]]*EulerPhi[i]}]]];
%t A116930 a[n_] := b[n, n][[2]];
%t A116930 Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Mar 13 2022, after _Alois P. Heinz_ *)
%Y A116930 Cf. A000010, A116929, A014153.
%K A116930 nonn
%O A116930 1,3
%A A116930 _Emeric Deutsch_, Feb 27 2006