A196087 Sum of all parts minus the total numbers of parts of all partitions of n.

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

Offset: 1

Author

Omar E. Pol, Nov 10 2011

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000041, A006128, A066186, A207034.

Programs

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-François Alcover, Oct 22 2015, after Alois P. Heinz *)

PARI

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

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

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

a(n) = Sum_{k=1..n-1} p(n+j,j), where p(n,j) is the number of partitions of j having exactly j parts. E.g., a(4) = p(5,1) + p(6,2) + p(7,3) = 1+3+4 = 8. - Gregory L. Simay, Aug 19 2022