cp's OEIS Frontend

A207382 Sum of the even-indexed parts of all partitions of n.

%I A207382 #35 Apr 22 2017 23:26:12
%S A207382 0,1,2,6,10,21,33,59,90,145,213,328,467,684,959,1361,1866,2588,3490,
%T A207382 4741,6311,8422,11067,14579,18941,24630,31703,40788,52019,66315,83891,
%U A207382 106034,133182,167045,208397,259637,321895,398498,491295,604725,741579,908008
%N A207382 Sum of the even-indexed parts of all partitions of n.
%C A207382 Also the sum of the floors of half the parts of all partitions of n, because the sum of one kind for a partition equals the sum of the other kind for the conjugate partition. Furthermore, this generalizes to taking m-th indices and dividing by m. - _George Beck_, Apr 15 2017
%H A207382 Alois P. Heinz, <a href="/A207382/b207382.txt">Table of n, a(n) for n = 1..1000</a>
%F A207382 a(n) = A066186(n) - A207381(n) = A207381(n) - A066897(n).
%e A207382 For n = 5, write the partitions of 5 and below write the sums of their even-indexed parts:
%e A207382 . 5
%e A207382 . 3+2
%e A207382 . 4+1
%e A207382 . 2+2+1
%e A207382 . 3+1+1
%e A207382 . 2+1+1+1
%e A207382 . 1+1+1+1+1
%e A207382 ------------
%e A207382 .   8 + 2   = 10
%e A207382 The sum of the even-indexed parts is 10, so a(5) = 10.
%e A207382 From _George Beck_, Apr 15 2017: (Start)
%e A207382 Alternatively, sum the floors of the parts divided by 2:
%e A207382 . 2
%e A207382 . 1+1
%e A207382 . 2+0
%e A207382 . 1+1+0
%e A207382 . 1+0+0
%e A207382 . 1+0+0+0
%e A207382 . 0+0+0+0+0
%e A207382 The sum is 10, so a(5) = 10. (End)
%p A207382 b:= proc(n, i) option remember; local g, h;
%p A207382       if n=0 then [1, 0$2]
%p A207382     elif i<1 then [0$3]
%p A207382     else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
%p A207382          [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
%p A207382       fi
%p A207382     end:
%p A207382 a:= n-> b(n,n)[2]:
%p A207382 seq (a(n), n=1..50); # _Alois P. Heinz_, Mar 12 2012
%t A207382 b[n_, i_] := b[n, i] = Module[{g, h}, Which[n==0, {1, 0, 0}, i<1, {0, 0, 0}, True, 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][[2]]; Table [a[n], {n, 1, 50}] (* _Jean-François Alcover_, Feb 03 2017, after _Alois P. Heinz_ *)
%t A207382 a[n_]:= Total@Flatten@Quotient[IntegerPartitions[n], 2];
%t A207382 Table [a[n], {n, 1, 50}] (* _George Beck_, Apr 15 2017 *)
%Y A207382 For more information see A206563.
%Y A207382 Cf. A066186, A066897, A066898, A181187, A194714, A206283, A207031, A207032, A207381.
%K A207382 nonn
%O A207382 1,3
%A A207382 _Omar E. Pol_, Feb 17 2012
%E A207382 More terms from _Alois P. Heinz_, Mar 12 2012