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A222044 Sum of smallest parts of all partitions of n into an odd number of parts.

%I A222044 #18 Jul 06 2019 16:54:17
%S A222044 0,1,2,4,5,8,11,15,19,28,35,47,61,80,102,136,168,218,276,350,437,556,
%T A222044 686,860,1063,1321,1620,2005,2443,2998,3649,4445,5377,6531,7863,9496,
%U A222044 11398,13694,16373,19603,23347,27834,33058,39259,46467,55020,64914,76599
%N A222044 Sum of smallest parts of all partitions of n into an odd number of parts.
%C A222044 a(n) + A222045(n) = A046746(n).
%C A222044 a(n) - A222045(n) = A222046(n).
%H A222044 Vaclav Kotesovec, <a href="/A222044/b222044.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A222044 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - _Vaclav Kotesovec_, Jul 06 2019
%e A222044 a(6) = 11: partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6], sum of smallest parts is 1+2+1+1+6 = 11.
%p A222044 b:= proc(n, i) option remember;
%p A222044       [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
%p A222044        `if`(n<i, [0, 0], (l-> [l[2], l[1]])(b(n-i, i))))
%p A222044     end:
%p A222044 a:= n-> b(n, n)[1]:
%p A222044 seq(a(n), n=0..60);
%t A222044 b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[n<i, {0, 0}, Reverse[b[n-i, i]]]]; a[n_] := b[n, n][[1]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 03 2017, translated from Maple *)
%t A222044 Table[Total[Min/@Select[IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,50}] (* _Harvey P. Dale_, Jul 05 2019 *)
%Y A222044 Cf. A046746, A211373, A222045, A222046, A222047, A222048, A222049.
%K A222044 nonn
%O A222044 0,3
%A A222044 _Alois P. Heinz_, Feb 06 2013