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A207382 Sum of the even-indexed parts of all partitions of n. 4
0, 1, 2, 6, 10, 21, 33, 59, 90, 145, 213, 328, 467, 684, 959, 1361, 1866, 2588, 3490, 4741, 6311, 8422, 11067, 14579, 18941, 24630, 31703, 40788, 52019, 66315, 83891, 106034, 133182, 167045, 208397, 259637, 321895, 398498, 491295, 604725, 741579, 908008 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

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

EXAMPLE

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

. 5

. 3+2

. 4+1

. 2+2+1

. 3+1+1

. 2+1+1+1

. 1+1+1+1+1

------------

.   8 + 2   = 10

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

From George Beck, Apr 15 2017: (Start)

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

. 2

. 1+1

. 2+0

. 1+1+0

. 1+0+0

. 1+0+0+0

. 0+0+0+0+0

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

MAPLE

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

      if n=0 then [1, 0$2]

    elif i<1 then [0$3]

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

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

      fi

    end:

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

seq (a(n), n=1..50); # Alois P. Heinz, Mar 12 2012

MATHEMATICA

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-Fran├žois Alcover, Feb 03 2017, after Alois P. Heinz *)

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

Table [a[n], {n, 1, 50}] (* George Beck, Apr 15 2017 *)

CROSSREFS

For more information see A206563.

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

Sequence in context: A125518 A083176 A103628 * A272952 A034450 A297185

Adjacent sequences:  A207379 A207380 A207381 * A207383 A207384 A207385

KEYWORD

nonn

AUTHOR

Omar E. Pol, Feb 17 2012

EXTENSIONS

More terms from Alois P. Heinz, Mar 12 2012

STATUS

approved

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Last modified February 20 07:41 EST 2020. Contains 332069 sequences. (Running on oeis4.)