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A103628 Total sum of parts of multiplicity 1 in all partitions of n.

%I A103628 #32 Jul 02 2025 16:02:02
%S A103628 0,1,2,6,10,21,33,59,89,145,212,325,463,680,948,1348,1845,2558,3446,
%T A103628 4681,6219,8306,10901,14352,18632,24230,31151,40077,51074,65088,82290,
%U A103628 103986,130517,163679,204078,254174,314975,389839,480369,591133,724600,886965
%N A103628 Total sum of parts of multiplicity 1 in all partitions of n.
%C A103628 Total number of parts of multiplicity 1 in all partitions of n is A024786(n+1).
%C A103628 Equals A000041 convolved with A026741. - _Gary W. Adamson_, Jun 11 2009
%H A103628 Alois P. Heinz, <a href="/A103628/b103628.txt">Table of n, a(n) for n = 0..1000</a>
%F A103628 G.f.: x*(1+x+x^2)/(1-x^2)^2 /Product_{k>0}(1-x^k).
%F A103628 a(n) = A066186(n) - A194544(n). - _Omar E. Pol_, Nov 20 2011
%F A103628 a(n) = 3*A014153(n)/4 - 3*A000070(n)/4 - A270143(n+1)/4 + A087787(n)/4. - _Vaclav Kotesovec_, Nov 05 2016
%F A103628 a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2) * (1 - Pi/(24*sqrt(6*n))). - _Vaclav Kotesovec_, Nov 05 2016
%e A103628 Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and a(4) = 0 + 2 + 0 + (1+3) + 4 = 10.
%p A103628 gf:=x*(1+x+x^2)/(1-x^2)^2/product((1-x^k), k=1..500): s:=series(gf, x, 100): for n from 0 to 60 do printf(`%d,`,coeff(s, x, n)) od: # _James Sellers_, Apr 22 2005
%p A103628 # second Maple program:
%p A103628 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A103628       `if`(i<1, [0, 0], add((l->`if`(j=1, [l[1],
%p A103628        l[2]+l[1]*i], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A103628     end:
%p A103628 a:= n-> b(n, n)[2]:
%p A103628 seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 03 2013
%t A103628 b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[3]]; a[0] = 0; Table[a[n], {n, 0, 50}]  (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)
%Y A103628 Cf. A026741. - _Gary W. Adamson_, Jun 11 2009
%Y A103628 Column k=1 of A222730. - _Alois P. Heinz_, Mar 03 2013
%K A103628 easy,nonn
%O A103628 0,3
%A A103628 _Vladeta Jovovic_, Mar 25 2005
%E A103628 More terms from _James Sellers_, Apr 22 2005