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

%I A138135 #29 Mar 05 2021 21:43:38
%S A138135 0,1,1,3,3,8,8,17,20,34,41,68,80,123,153,219,271,382,469,642,795,1055,
%T A138135 1305,1713,2102,2713,3336,4241,5190,6545,7968,9950,12090,14953,18104,
%U A138135 22255,26821,32752,39371,47774,57220,69104
%N A138135 Number of parts > 1 in the last section of the set of partitions of n.
%C A138135 Also first differences of A096541. For more information see A135010.
%H A138135 Alois P. Heinz, <a href="/A138135/b138135.txt">Table of n, a(n) for n = 1..1000</a>
%F A138135 a(n) = A096541(n)-A096541(n-1) = A138137(n)-A000041(n-1) = A006128(n)-A006128(n-1)-A000041(n-1).
%F A138135 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
%F A138135 G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k) / Product_{j>=2} (1 - x^j). - _Ilya Gutkovskiy_, Mar 05 2021
%p A138135 b:= proc(n, i) option remember; local f, g;
%p A138135       if n=0 or i=1 then [1, 0]
%p A138135     else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
%p A138135          [f[1]+g[1], f[2]+g[2]+`if`(i>1, g[1], 0)]
%p A138135       fi
%p A138135     end:
%p A138135 a:= n-> b(n, n)[2]-b(n-1, n-1)[2]:
%p A138135 seq (a(n), n=1..60); # _Alois P. Heinz_, Apr 04 2012
%t A138135 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 *)
%t A138135 Table[Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 42}]  (* _Robert Price_, May 01 2020 *)
%o A138135 (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
%Y A138135 Zero together with the column k=2 of A207031.
%Y A138135 Cf. A000041, A006128, A096541, A135010, A138121, A138137.
%K A138135 nonn
%O A138135 1,4
%A A138135 _Omar E. Pol_, Mar 30 2008