A222044 - OEIS

%I #18 Jul 06 2019 16:54:17

%S 0,1,2,4,5,8,11,15,19,28,35,47,61,80,102,136,168,218,276,350,437,556,

%T 686,860,1063,1321,1620,2005,2443,2998,3649,4445,5377,6531,7863,9496,

%U 11398,13694,16373,19603,23347,27834,33058,39259,46467,55020,64914,76599

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

%C a(n) + A222045(n) = A046746(n).

%C a(n) - A222045(n) = A222046(n).

%H 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 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - _Vaclav Kotesovec_, Jul 06 2019

%e 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 b:= proc(n, i) option remember;

%p       [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+

%p        `if`(n<i, [0, 0], (l-> [l[2], l[1]])(b(n-i, i))))

%p     end:

%p a:= n-> b(n, n)[1]:

%p seq(a(n), n=0..60);

%t 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 Table[Total[Min/@Select[IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,50}] (* _Harvey P. Dale_, Jul 05 2019 *)

%Y Cf. A046746, A211373, A222045, A222046, A222047, A222048, A222049.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 06 2013