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A206434 Total number of even parts in the last section of the set of partitions of n.

%I A206434 #31 Apr 30 2023 22:20:19
%S A206434 0,1,0,3,1,6,4,13,10,24,23,46,46,81,88,143,159,242,278,404,470,657,
%T A206434 776,1057,1251,1663,1984,2587,3089,3967,4742,6012,7184,9001,10753,
%U A206434 13351,15917,19594,23335,28514,33883,41140,48787,58894,69691,83680,98809,118101
%N A206434 Total number of even parts in the last section of the set of partitions of n.
%C A206434 From _Omar E. Pol_, Apr 07 2023: (Start)
%C A206434 Convolution of A002865 and A183063.
%C A206434 a(n) is also the total number of even divisors of the terms in the n-th row of the triangle A336811.
%C A206434 a(n) is also the number of even terms in the n-th row of the triangle A207378.
%C A206434 a(n) is also the number of even terms in the n-th row of the triangle A336812. (End)
%H A206434 Alois P. Heinz, <a href="/A206434/b206434.txt">Table of n, a(n) for n = 1..1000</a>
%F A206434 G.f.: (Sum_{i>0} (x^(2*i)-x^(2*i+1))/(1-x^(2*i)))/Product_{i>0} (1-x^i). - _Alois P. Heinz_, Mar 23 2012
%p A206434 b:= proc(n, i) option remember; local f, g;
%p A206434       if n=0 or i=1 then [1, 0]
%p A206434     else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
%p A206434          [f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]]
%p A206434       fi
%p A206434     end:
%p A206434 a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
%p A206434 seq (a(n), n=1..50);  # _Alois P. Heinz_, Mar 22 2012
%t A206434 b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + Mod[i+1, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[ a[n], {n, 1, 50}] (* _Jean-François Alcover_, Feb 16 2017, after _Alois P. Heinz_ *)
%Y A206434 Partial sums give A066898.
%Y A206434 Cf. A002865, A006128, A135010, A138121, A138137, A182703, A183063, A206433, A206435, A206436, A207378, A336811, A336812.
%K A206434 nonn
%O A206434 1,4
%A A206434 _Omar E. Pol_, Feb 12 2012
%E A206434 More terms from _Alois P. Heinz_, Mar 22 2012