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A337165 Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A337165 #51 Feb 07 2021 07:04:26
%S A337165 1,0,1,0,0,1,0,0,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,0,3,0,0,1,0,0,0,0,4,0,
%T A337165 0,1,0,0,1,0,0,5,0,0,1,0,1,0,3,0,0,6,0,0,1,0,0,2,0,6,0,0,7,0,0,1,0,0,
%U A337165 0,3,0,10,0,0,8,0,0,1,0,0,0,1,4,0,15,0,0,9,0,0,1
%N A337165 Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A337165 Alois P. Heinz, <a href="/A337165/b337165.txt">Rows n = 0..350, flattened</a>
%F A337165 G.f. of column k: (Sum_{j>=1} x^(j^2))^k.
%F A337165 Sum_{k=0..n} k * T(n,k) = A281704(n).
%F A337165 Sum_{k=0..n} (-1)^k * T(n,k) = A317665(n).
%e A337165 Triangle T(n,k) begins:
%e A337165   1;
%e A337165   0, 1;
%e A337165   0, 0, 1;
%e A337165   0, 0, 0, 1;
%e A337165   0, 1, 0, 0, 1;
%e A337165   0, 0, 2, 0, 0,  1;
%e A337165   0, 0, 0, 3, 0,  0,  1;
%e A337165   0, 0, 0, 0, 4,  0,  0, 1;
%e A337165   0, 0, 1, 0, 0,  5,  0, 0, 1;
%e A337165   0, 1, 0, 3, 0,  0,  6, 0, 0, 1;
%e A337165   0, 0, 2, 0, 6,  0,  0, 7, 0, 0, 1;
%e A337165   0, 0, 0, 3, 0, 10,  0, 0, 8, 0, 0, 1;
%e A337165   0, 0, 0, 1, 4,  0, 15, 0, 0, 9, 0, 0, 1;
%e A337165   ...
%p A337165 b:= proc(n) option remember; `if`(n=0, 1, add((s->
%p A337165      `if`(s>n, 0, expand(x*b(n-s))))(j^2), j=1..isqrt(n)))
%p A337165     end:
%p A337165 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p A337165 seq(T(n), n=0..14);
%t A337165 b[n_] := b[n] = If[n == 0, 1, Sum[With[{s = j^2},
%t A337165      If[s>n, 0, Expand[x*b[n - s]]]], {j, 1, Sqrt[n]}]];
%t A337165 T[n_] := CoefficientList[b[n], x];
%t A337165 T /@ Range[0, 14] // Flatten (* _Jean-François Alcover_, Feb 07 2021, after _Alois P. Heinz_ *)
%Y A337165 Columns k=0-10 give: A000007, A010052, A063725, A063691, A063730, A340481, A340905, A340906, A340915, A340946, A340947.
%Y A337165 Row sums give A006456.
%Y A337165 T(2n,n) gives A338464.
%Y A337165 Main diagonal gives A000012.
%Y A337165 Cf. A000290, A281704, A317665, A341040.
%K A337165 nonn,tabl
%O A337165 0,18
%A A337165 _Alois P. Heinz_, Feb 03 2021