A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

0, 1, 2, 5, 7, 16, 20, 39, 52, 86, 113, 184, 232, 353, 462, 661, 851, 1202, 1526, 2098, 2670, 3565, 4514, 5967, 7473, 9715, 12162, 15583, 19373, 24625, 30410, 38274, 47112, 58725, 71951, 89129, 108599, 133612, 162259, 198346, 239825, 291718, 351269, 425102

Offset

1,3

Links

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

Formula

a(n) = A138879(n) - A138137(n) = A138880(n) - A138135(n). - Omar E. Pol, Apr 21 2012

G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^2 / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

Example

For n = 7 the last section of the set of partitions of 7 looks like this:
.
.        (. . . . . . 7)
.        (. . . 4 . . 3)
.        (. . . . 5 . 2)
.        (. . 3 . 2 . 2)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.
The sum of all parts = 7+4+3+5+2+3+2+2+1*11 = 39, on the other hand the total number of parts is 1+2+2+3+1*11 = 19, so a(7) = 39 - 19 = 20. Note that the number of dots in the picture is also equal to a(7) = 6+5+5+4 = 20.

Maple

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]*(i-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Feb 20 2012

Mathematica

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n==0, {1, 0}, i<2, {0, 0}, i>n , b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]*(i-1)}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 13 2015, after Alois P. Heinz *)

Crossrefs

Row sums of triangle A207034. Partial sums give A196087.

Cf. A006128, A066186, A135010, A138121, A138135, A138137, A138879, A138880, A187219, A194548, A207038.

Sequence in context: A285290 A032141 A032045 * A322440 A067580 A325210

Adjacent sequences: A207032 A207033 A207034 * A207036 A207037 A207038

Keyword

nonn

Author

Omar E. Pol, Feb 20 2012

Extensions

More terms from Alois P. Heinz, Feb 20 2012

Status

approved