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

1, 3, 7, 14, 25, 45, 72, 117, 180, 275, 403, 596, 846, 1206, 1681, 2335, 3183, 4342, 5820, 7799, 10321, 13622, 17798, 23221, 30009, 38706, 49567, 63316, 80366, 101805, 128211, 161134, 201537, 251495, 312508, 387535, 478674, 590072, 724920, 888795, 1086324

Offset

1,2

Comments

For more information see A206563.

Links

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

Formula

a(n) = A066186(n) - A207382(n) = A066897(n) + A207382(n).

Example

For n = 5, write the partitions of 5 and below write the sums of their odd-indexed parts:
.    5
.    3+2
.    4+1
.    2+2+1
.    3+1+1
.    2+1+1+1
.    1+1+1+1+1
.  ------------
.   20 + 4 + 1 = 25
The total sum of the odd-indexed parts is 25 so a(5) = 25.

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)[3]:
seq(a(n), n=1..50); # Alois P. Heinz, Mar 12 2012

Mathematica

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 *)

Crossrefs

Cf. A066186, A066897, A066898, A181187, A194714, A206283, A206563, A207031, A207032, A207382.

Sequence in context: A265252 A193911 A206417 * A343017 A008646 A036830

Adjacent sequences: A207378 A207379 A207380 * A207382 A207383 A207384

Keyword

nonn

Author

Omar E. Pol, Feb 17 2012

Extensions

More terms from Alois P. Heinz, Mar 12 2012

Status

approved