A138135 Number of parts > 1 in the last section of the set of partitions of n.

0, 1, 1, 3, 3, 8, 8, 17, 20, 34, 41, 68, 80, 123, 153, 219, 271, 382, 469, 642, 795, 1055, 1305, 1713, 2102, 2713, 3336, 4241, 5190, 6545, 7968, 9950, 12090, 14953, 18104, 22255, 26821, 32752, 39371, 47774, 57220, 69104

Offset

1,4

Comments

Also first differences of A096541. For more information see A135010.

Links

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

Formula

a(n) = A096541(n)-A096541(n-1) = A138137(n)-A000041(n-1) = A006128(n)-A006128(n-1)-A000041(n-1).

a(n) ~ exp(Pi*sqrt(2*n/3))*(2*gamma - 2 + log(6*n/Pi^2))/(8*sqrt(3)*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) / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

Maple

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]+`if`(i>1, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]-b(n-1, n-1)[2]:
seq (a(n), n=1..60); # Alois P. Heinz, Apr 04 2012

Mathematica

a[n_] := DivisorSigma[0, n] - 1 + Sum[(DivisorSigma[0, k] - 1)*(PartitionsP[n - k] - PartitionsP[n - k - 1]), {k, 1, n - 1}]; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Jan 14 2013, from 1st formula *)
Table[Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 42}]  (* Robert Price, May 01 2020 *)

Prog

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

Crossrefs

Zero together with the column k=2 of A207031.

Cf. A000041, A006128, A096541, A135010, A138121, A138137.

Sequence in context: A205977 A363725 A238623 * A113166 A126872 A336102

Adjacent sequences: A138132 A138133 A138134 * A138136 A138137 A138138

Keyword

nonn

Author

Omar E. Pol, Mar 30 2008

Status

approved