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A264405 Triangle read by rows: T(n,k) is the number of integer partitions of n having k repeated parts (each occurrence is counted).

%I A264405 #13 Jan 23 2016 08:34:47
%S A264405 1,1,0,1,0,1,2,0,0,1,2,0,2,0,1,3,0,2,1,0,1,4,0,2,2,2,0,1,5,0,4,2,1,2,
%T A264405 0,1,6,0,6,2,3,2,2,0,1,8,0,7,4,4,2,2,2,0,1,10,0,8,6,6,4,3,2,2,0,1,12,
%U A264405 0,13,6,6,8,3,3,2,2,0,1,15,0,15,9,11,6,9,4,3,2,2,0,1,18,0,21,10,13,12,7,8,4,3,2,2,0,1
%N A264405 Triangle read by rows: T(n,k) is the number of integer partitions of n having k repeated parts (each occurrence is counted).
%C A264405 Compare with A264052 where only one occurrence of a repeated part is counted.
%C A264405 Sum of entries in row n = number of partitions of n = A000041(n).
%C A264405 Sum_{k>=0} k*T(n,k) = A194452(n).
%H A264405 Alois P. Heinz, <a href="/A264405/b264405.txt">Rows n = 0..200, flattened</a>
%F A264405 G.f.: G(t,x) = Product_{j>=1}(1 + x^j + t^2*x^{2j}/(1 - tx^j)).
%e A264405 T(4,2) = 2 because each of the partitions [2,2] and [2,1,1] have 2 repeated parts, while [4], [3,1], [1,1,1,1] have 0 or 4 repeated parts.
%e A264405 Triangle starts:
%e A264405   1;
%e A264405   1, 0;
%e A264405   1, 0, 1;
%e A264405   2, 0, 0, 1;
%e A264405   2, 0, 2, 0, 1;
%e A264405   3, 0, 2, 1, 0, 1;
%p A264405 g := product(1+x^j+t^2*x^(2*j)/(1-t*x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): 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 A264405 # second Maple program:
%p A264405 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A264405       add(expand(b(n-i*j, i-1)*`if`(j>1, x^j, 1)), j=0..n/i)))
%p A264405     end:
%p A264405 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A264405 seq(T(n), n=0..14);  # _Alois P. Heinz_, Dec 07 2015
%t A264405 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^j, 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, 14}] // Flatten (* _Jean-François Alcover_, Jan 23 2016, after _Alois P. Heinz_ *)
%Y A264405 Cf. A000009, A000041, A194452, A264052.
%K A264405 nonn,tabl
%O A264405 0,7
%A A264405 _Emeric Deutsch_, Dec 07 2015