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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.

%I A096541 #59 Mar 28 2023 14:01:37
%S A096541 0,0,1,2,5,8,16,24,41,61,95,136,204,284,407,560,779,1050,1432,1901,
%T A096541 2543,3338,4393,5698,7411,9513,12226,15562,19803,24993,31538,39506,
%U A096541 49456,61546,76499,94603,116858,143679,176431,215802,263576,320796,389900
%N 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.
%C A096541 Also column 2 of A181187. - _Omar E. Pol_, Feb 18 2012
%C A096541 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
%H A096541 Alois P. Heinz, <a href="/A096541/b096541.txt">Table of n, a(n) for n = 0..1000</a>
%F A096541 a(n) = A093694(n) - A000070(n).
%F A096541 a(n) = Sum_{k=1..n} (tau(k)-1)*numbpart(n-k). - _Vladeta Jovovic_, Jun 26 2004
%F A096541 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
%F A096541 a(n) = Sum_{i=1..floor(n/2)} A066633(n-i,i). - _George Beck_, Feb 15 2020
%F A096541 G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k) / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Mar 05 2021
%e A096541 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.
%p A096541 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;
%p A096541 # second Maple program:
%p A096541 b:= proc(n, i) option remember; local f, g;
%p A096541       if n=0 or i=1 then [1, 0]
%p A096541     else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
%p A096541          [f[1]+g[1], f[2]+g[2]+g[1]]
%p A096541       fi
%p A096541     end:
%p A096541 a:= n-> b(n, n)[2]:
%p A096541 seq(a(n), n=0..60); # _Alois P. Heinz_, Apr 04 2012
%t A096541 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 *)
%t A096541 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_ *)
%o A096541 (PARI) a(n)=sum(k=1,n,(numdiv(k)-1)*numbpart(n-k)) \\ _Charles R Greathouse IV_, Jan 14 2013
%Y A096541 Cf. A066633, A093694, A093695, A094533, A006128, A339011.
%K A096541 nonn
%O A096541 0,4
%A A096541 _Thomas Wieder_, Jun 24 2004
%E A096541 More terms from _Robert G. Wilson v_, Jun 30 2004