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A206436 Total sum of even parts in the last section of the set of partitions of n.

%I A206436 #30 Apr 29 2023 16:16:59
%S A206436 0,2,0,8,2,18,10,42,28,80,70,162,148,290,300,530,562,918,1020,1570,
%T A206436 1780,2602,3022,4286,4992,6858,8110,10872,12888,16962,20178,26134,
%U A206436 31138,39728,47412,59848,71312,89072,106176,131440,156400,192164,228330,278616,330502
%N A206436 Total sum of even parts in the last section of the set of partitions of n.
%C A206436 Also total sum of even parts in the partitions of n that do not contain 1 as a part.
%C A206436 From _Omar E. Pol_, Apr 09 2023: (Start)
%C A206436 Convolution of A002865 and A146076.
%C A206436 a(n) is also the total sum of even divisors of the terms in the n-th row of the triangle A336811.
%C A206436 a(n) is also the sum of even terms in the n-th row of the triangle A207378.
%C A206436 a(n) is also the sum of even terms in the n-th row of the triangle A336812. (End)
%H A206436 Vaclav Kotesovec, <a href="/A206436/b206436.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A206436 G.f.: (Sum_{i>0} 2*i*x^(2*i)*(1-x)/(1-x^(2*i))) / Product_{i>0} (1-x^i). - _Alois P. Heinz_, Mar 16 2012
%F A206436 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2*n)). - _Vaclav Kotesovec_, May 29 2018
%p A206436 b:= proc(n, i) option remember; local g, h;
%p A206436       if n=0 then [1, 0]
%p A206436     elif i<1 then [0, 0]
%p A206436     else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
%p A206436          [g[1]+h[1], g[2]+h[2] +((i+1) mod 2)*h[1]*i]
%p A206436       fi
%p A206436     end:
%p A206436 a:= n-> b(n, n)[2] -`if`(n=1, 0, b(n-1, n-1)[2]):
%p A206436 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 16 2012
%t A206436 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+1, 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 A206436 Partial sums give A066966.
%Y A206436 Cf. A002865, A135010, A138121, A138879, A146076, A206433, A206434, A206435, A207378, A336811, A336812.
%K A206436 nonn
%O A206436 1,2
%A A206436 _Omar E. Pol_, Feb 12 2012
%E A206436 More terms from _Alois P. Heinz_, Mar 16 2012