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A196087 Sum of all parts minus the total numbers of parts of all partitions of n. 8
0, 1, 3, 8, 15, 31, 51, 90, 142, 228, 341, 525, 757, 1110, 1572, 2233, 3084, 4286, 5812, 7910, 10580, 14145, 18659, 24626, 32099, 41814, 53976, 69559, 88932, 113557, 143967, 182241, 229353, 288078, 360029, 449158, 557757, 691369, 853628, 1051974 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Also sum of parts of all partitions of n except the largest parts of the partitions. - Omar E. Pol, Feb 16 2012

Equals column 1 of A161224. - Omar E. Pol, Feb 26 2012

Partial sums of A207035. - Omar E. Pol, Apr 22 2012

LINKS

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

FORMULA

a(n) = n*A000041(n) - A006128(n) = A066186(n) - A006128(n).

a(n) = A207038(A000041(n)). - Omar E. Pol, Apr 21 2012

a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)) * (1 - (3 + 6*gamma + Pi^2/24 + 3*log(6*n/Pi^2))/(Pi*sqrt(6*n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 24 2016

EXAMPLE

For n = 4 the five partitions of 4 are: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. The sum of all parts is 4+3+1+2+2+2+1+1+1+1+1+1 = 20. The sum of all parts is also the product n*p(n) = 4*5 = 20, where p(n) = A000041(n) is the number of partitions of n. On the other hand the number of parts in all partitions of 4 is equal to 12, so a(4) = 20-12 = 8.

MAPLE

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

      if n=0 then [1, 0]

    elif i<1 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<1, {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-Fran├žois Alcover, Oct 22 2015, after Alois P. Heinz *)

PROG

(PARI) a(n) = n*numbpart(n) - sum(m=1, n, numdiv(m)*numbpart(n-m)); \\ Michel Marcus, Oct 22 2015

CROSSREFS

Cf. A000041, A006128, A066186, A207034.

Sequence in context: A317252 A135350 A068038 * A295735 A309052 A328858

Adjacent sequences:  A196084 A196085 A196086 * A196088 A196089 A196090

KEYWORD

nonn

AUTHOR

Omar E. Pol, Nov 10 2011

STATUS

approved

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Last modified December 6 06:59 EST 2019. Contains 329784 sequences. (Running on oeis4.)