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A207381 Total sum of the odd-indexed parts of all partitions of n.

%I A207381 #29 Dec 09 2016 10:14:42
%S A207381 1,3,7,14,25,45,72,117,180,275,403,596,846,1206,1681,2335,3183,4342,
%T A207381 5820,7799,10321,13622,17798,23221,30009,38706,49567,63316,80366,
%U A207381 101805,128211,161134,201537,251495,312508,387535,478674,590072,724920,888795,1086324
%N A207381 Total sum of the odd-indexed parts of all partitions of n.
%C A207381 For more information see A206563.
%H A207381 Alois P. Heinz, <a href="/A207381/b207381.txt">Table of n, a(n) for n = 1..1000</a>
%F A207381 a(n) = A066186(n) - A207382(n) = A066897(n) + A207382(n).
%e A207381 For n = 5, write the partitions of 5 and below write the sums of their odd-indexed parts:
%e A207381 .    5
%e A207381 .    3+2
%e A207381 .    4+1
%e A207381 .    2+2+1
%e A207381 .    3+1+1
%e A207381 .    2+1+1+1
%e A207381 .    1+1+1+1+1
%e A207381 .  ------------
%e A207381 .   20 + 4 + 1 = 25
%e A207381 The total sum of the odd-indexed parts is 25 so a(5) = 25.
%p A207381 b:= proc(n, i) option remember; local g, h;
%p A207381       if n=0 then [1, 0$2]
%p A207381     elif i<1 then [0$3]
%p A207381     else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
%p A207381          [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
%p A207381       fi
%p A207381     end:
%p A207381 a:= n-> b(n,n)[3]:
%p A207381 seq(a(n), n=1..50); # _Alois P. Heinz_, Mar 12 2012
%t A207381 b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0 , {1, 0, 0}, If[i < 1, {0, 0, 0},  g = b[n, i - 1]; h = If[i > n, {0, 0, 0}, b[n - i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[3]], g[[3]] + h[[2]] + i*h[[1]]}]]]; a[n_] := b[n, n][[3]]; Table [a[n], {n, 1, 50}] (* _Jean-François Alcover_, Dec 09 2016 after _Alois P. Heinz_ *)
%Y A207381 Cf. A066186, A066897, A066898, A181187, A194714, A206283, A206563, A207031, A207032, A207382.
%K A207381 nonn
%O A207381 1,2
%A A207381 _Omar E. Pol_, Feb 17 2012
%E A207381 More terms from _Alois P. Heinz_, Mar 12 2012