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Sum of the even-indexed parts of all partitions of n.
4

%I #35 Apr 22 2017 23:26:12

%S 0,1,2,6,10,21,33,59,90,145,213,328,467,684,959,1361,1866,2588,3490,

%T 4741,6311,8422,11067,14579,18941,24630,31703,40788,52019,66315,83891,

%U 106034,133182,167045,208397,259637,321895,398498,491295,604725,741579,908008

%N Sum of the even-indexed parts of all partitions of n.

%C 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 Alois P. Heinz, <a href="/A207382/b207382.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A066186(n) - A207381(n) = A207381(n) - A066897(n).

%e For n = 5, write the partitions of 5 and below write the sums of their even-indexed parts:

%e . 5

%e . 3+2

%e . 4+1

%e . 2+2+1

%e . 3+1+1

%e . 2+1+1+1

%e . 1+1+1+1+1

%e ------------

%e . 8 + 2 = 10

%e The sum of the even-indexed parts is 10, so a(5) = 10.

%e From _George Beck_, Apr 15 2017: (Start)

%e Alternatively, sum the floors of the parts divided by 2:

%e . 2

%e . 1+1

%e . 2+0

%e . 1+1+0

%e . 1+0+0

%e . 1+0+0+0

%e . 0+0+0+0+0

%e The sum is 10, so a(5) = 10. (End)

%p b:= proc(n, i) option remember; local g, h;

%p if n=0 then [1, 0$2]

%p elif i<1 then [0$3]

%p else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));

%p [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]

%p fi

%p end:

%p a:= n-> b(n,n)[2]:

%p seq (a(n), n=1..50); # _Alois P. Heinz_, Mar 12 2012

%t 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 a[n_]:= Total@Flatten@Quotient[IntegerPartitions[n], 2];

%t Table [a[n], {n, 1, 50}] (* _George Beck_, Apr 15 2017 *)

%Y For more information see A206563.

%Y Cf. A066186, A066897, A066898, A181187, A194714, A206283, A207031, A207032, A207381.

%K nonn

%O 1,3

%A _Omar E. Pol_, Feb 17 2012

%E More terms from _Alois P. Heinz_, Mar 12 2012