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A264403 Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity 1 is equal to k (0<=k<=n).

%I A264403 #16 Jan 29 2016 10:50:29
%S A264403 1,0,1,1,0,1,1,0,0,2,2,0,1,0,2,1,1,1,1,0,3,4,0,1,1,1,0,4,2,2,1,2,1,2,
%T A264403 0,5,6,0,2,1,3,2,2,0,6,5,2,1,4,1,4,2,3,0,8,9,1,3,2,5,2,4,3,3,0,10,7,3,
%U A264403 3,6,2,7,2,6,3,5,0,12,16,0,4,4,7,3,8,3,7,5,5,0,15,11,6,4,8,5,9,3,12,3,10,5,7,0,18
%N A264403 Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity 1 is equal to k (0<=k<=n).
%C A264403 Row n contains n+1 entries (n>=0).
%C A264403 Row sums yield the partition numbers (A000041).
%C A264403 T(n,0) = A007690(n).
%C A264403 T(n,n) = A000009(n).
%C A264403 Sum_{k>=0} k*T(n,k) = A103628(n).
%H A264403 Alois P. Heinz, <a href="/A264403/b264403.txt">Rows n = 0..200, flattened</a>
%F A264403 G.f.: G(t,x) = Product_{j>=1} (1+t^j*x^j + x^{2*j}/(1 - x^j)).
%e A264403 T(7,5) = 2 because we have [3,2,1,1] and [5,1,1].
%e A264403 Triangle starts:
%e A264403 1;
%e A264403 0,1;
%e A264403 1,0,1;
%e A264403 1,0,0,2;
%e A264403 2,0,1,0,2;
%p A264403 g := product(1+t^j*x^j+x^(2*j)/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 25)): for n from 0 to 20 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%p A264403 # second Maple program:
%p A264403 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A264403       `if`(i<1, 0, add(expand(b(n-i*j, i-1)*
%p A264403       `if`(j=1, x^i, 1)), j=0..n/i)))
%p A264403     end:
%p A264403 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A264403 seq(T(n), n=0..15);  # _Alois P. Heinz_, Nov 27 2015
%t A264403 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[b[n-i*j, i-1]*If[j == 1, x^i, 1]], {j, 0, n/i}]]]; T[n_] := Function[p,Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Jan 29 2016, after _Alois P. Heinz_ *)
%Y A264403 Cf. A000009, A000041, A007690, A103628.
%K A264403 nonn,look,tabl
%O A264403 0,10
%A A264403 _Emeric Deutsch_, Nov 27 2015