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A206435 Total sum of odd parts in the last section of the set of partitions of n.

%I A206435 #31 Apr 30 2023 18:55:25
%S A206435 1,1,5,3,13,13,29,29,66,70,126,146,241,287,450,526,791,963,1360,1660,
%T A206435 2312,2810,3799,4649,6158,7528,9824,11962,15393,18773,23804,28932,
%U A206435 36413,44093,54953,66419,82085,98929,121469,145865,177983,213241,258585,308861
%N A206435 Total sum of odd parts in the last section of the set of partitions of n.
%C A206435 From _Omar E. Pol_, Apr 09 2023: (Start)
%C A206435 Convolution of A002865 and A000593.
%C A206435 a(n) is also the total sum of odd divisors of the terms in the n-th row of the triangle A336811.
%C A206435 a(n) is also the sum of odd terms in the n-th row of the triangle A207378.
%C A206435 a(n) is also the sum of odd terms in the n-th row of the triangle A336812. (End)
%H A206435 Vaclav Kotesovec, <a href="/A206435/b206435.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A206435 G.f.: (Sum_{i>=0} (2*i+1)*x^(2*i)*(1-x)/(1-x^(2*i+1))) / Product_{j>0} (1-x^j). - _Alois P. Heinz_, Mar 16 2012
%F A206435 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2*n)). - _Vaclav Kotesovec_, May 29 2018
%p A206435 b:= proc(n, i) option remember; local g, h;
%p A206435       if n=0 then [1, 0]
%p A206435     elif i<1 then [0, 0]
%p A206435     else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
%p A206435          [g[1]+h[1], g[2]+h[2] +(i mod 2)*h[1]*i]
%p A206435       fi
%p A206435     end:
%p A206435 a:= n-> b(n, n)[2] -`if`(n=1, 0, b(n-1, n-1)[2]):
%p A206435 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 16 2012
%t A206435 b[n_, i_] := b[n, i] = Module[{g, h}, Which[n == 0, {1, 0}, i < 1, {0, 0}, True, g = b[n, i-1]; h = If[i > n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + Mod[i, 2]*h[[1]]*i}]]; a[n_] := b[n, n][[2]] - If[n == 1, 0, b[n-1, n-1][[2]]]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Feb 16 2017, after _Alois P. Heinz_ *)
%Y A206435 Partial sums give A066967.
%Y A206435 Cf. A000593, A002865, A135010, A138121, A138879, A206433, A206434, A206436, A207378, A336811, A336812.
%K A206435 nonn
%O A206435 1,3
%A A206435 _Omar E. Pol_, Feb 12 2012
%E A206435 More terms from _Alois P. Heinz_, Mar 16 2012