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A222045 Sum of smallest parts of all partitions of n into an even number of parts.

%I A222045 #16 Jul 06 2019 17:07:06
%S A222045 0,0,1,1,4,4,9,10,19,21,34,40,62,72,103,124,173,207,279,337,445,538,
%T A222045 694,842,1077,1299,1634,1977,2464,2969,3669,4411,5410,6488,7896,9447,
%U A222045 11442,13640,16421,19536,23411,27761,33124,39174,46554,54915,65008,76485,90258
%N A222045 Sum of smallest parts of all partitions of n into an even number of parts.
%C A222045 A222044(n) + a(n) = A046746(n).
%C A222045 A222044(n) - a(n) = A222046(n).
%H A222045 Vaclav Kotesovec, <a href="/A222045/b222045.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A222045 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - _Vaclav Kotesovec_, Jul 06 2019
%e A222045 a(6) = 9: partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1], sum of smallest parts is 1+1+1+3+2+1 = 9.
%p A222045 b:= proc(n, i) option remember;
%p A222045       [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
%p A222045        `if`(n<i, [0, 0], (l-> [l[2], l[1]])(b(n-i, i))))
%p A222045     end:
%p A222045 a:= n-> b(n, n)[2]:
%p A222045 seq(a(n), n=0..60);
%t A222045 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][[2]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 03 2017, translated from Maple *)
%Y A222045 Cf. A046746, A211373, A222044, A222046, A222047, A222048, A222049.
%K A222045 nonn
%O A222045 0,5
%A A222045 _Alois P. Heinz_, Feb 06 2013