A096541 Number of parts unequal to 1 in all partitions of the integer n. Also the difference between the labeled and the unlabeled case of one-element transitions from the partitions of n to the partitions of n+1.

0, 0, 1, 2, 5, 8, 16, 24, 41, 61, 95, 136, 204, 284, 407, 560, 779, 1050, 1432, 1901, 2543, 3338, 4393, 5698, 7411, 9513, 12226, 15562, 19803, 24993, 31538, 39506, 49456, 61546, 76499, 94603, 116858, 143679, 176431, 215802, 263576, 320796, 389900

Offset: 0

Author

Thomas Wieder, Jun 24 2004

Keywords

• nonn

Comments

Also column 2 of A181187. - Omar E. Pol, Feb 18 2012

Sum over all partitions of n of the difference between the number of parts and the number of distinct parts. - Alois P. Heinz, Nov 18 2020

Examples

			The partitions of n=5 are [11111], [1112], [113], [122], [23], [14], [5] and they contain 0 + 1 + 1 + 2 + 2 + 1 + 1 = 8 = A096541(5) parts unequal to 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A066633, A093694, A093695, A094533, A006128, A339011.

Programs

Maple

main := proc(n::integer) local a,ndxp,ndxprt,ListOfPartitions,iverbose; with(combinat): ListOfPartitions:=partition(n); a:=0; for ndxp from 1 to nops(ListOfPartitions) do for ndxprt from 1 to nops(ListOfPartitions[ndxp]) do if op(ndxprt,ListOfPartitions[ndxp]) <> 1 then a := a + 1; fi; end do; end do; print("n, a(n):",n,a); end proc;
# second Maple program:
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60); # Alois P. Heinz, Apr 04 2012

Mathematica

f[n_] := Block[{l = Sort[ Flatten[ IntegerPartitions[n]]]}, Length[l] - Count[l, 1]]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, Jun 30 2004 *)
a[n_] := Sum[(DivisorSigma[0, k] - 1)*PartitionsP[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jan 14 2013, after Vladeta Jovovic *)

PARI

a(n)=sum(k=1,n,(numdiv(k)-1)*numbpart(n-k)) \\ Charles R Greathouse IV, Jan 14 2013

Formula

a(n) = A093694(n) - A000070(n).

a(n) = Sum_{k=1..n} (tau(k)-1)*numbpart(n-k). - Vladeta Jovovic, Jun 26 2004

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

a(n) = Sum_{i=1..floor(n/2)} A066633(n-i,i). - George Beck, Feb 15 2020

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

Extensions

More terms from Robert G. Wilson v, Jun 30 2004