A337165 - OEIS

%I #51 Feb 07 2021 07:04:26

%S 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 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 0,3,0,10,0,0,8,0,0,1,0,0,0,1,4,0,15,0,0,9,0,0,1

%N 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 Alois P. Heinz, <a href="/A337165/b337165.txt">Rows n = 0..350, flattened</a>

%F G.f. of column k: (Sum_{j>=1} x^(j^2))^k.

%F Sum_{k=0..n} k * T(n,k) = A281704(n).

%F Sum_{k=0..n} (-1)^k * T(n,k) = A317665(n).

%e Triangle T(n,k) begins:

%e   1;

%e   0, 1;

%e   0, 0, 1;

%e   0, 0, 0, 1;

%e   0, 1, 0, 0, 1;

%e   0, 0, 2, 0, 0,  1;

%e   0, 0, 0, 3, 0,  0,  1;

%e   0, 0, 0, 0, 4,  0,  0, 1;

%e   0, 0, 1, 0, 0,  5,  0, 0, 1;

%e   0, 1, 0, 3, 0,  0,  6, 0, 0, 1;

%e   0, 0, 2, 0, 6,  0,  0, 7, 0, 0, 1;

%e   0, 0, 0, 3, 0, 10,  0, 0, 8, 0, 0, 1;

%e   0, 0, 0, 1, 4,  0, 15, 0, 0, 9, 0, 0, 1;

%e   ...

%p b:= proc(n) option remember; `if`(n=0, 1, add((s->

%p      `if`(s>n, 0, expand(x*b(n-s))))(j^2), j=1..isqrt(n)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):

%p seq(T(n), n=0..14);

%t b[n_] := b[n] = If[n == 0, 1, Sum[With[{s = j^2},

%t      If[s>n, 0, Expand[x*b[n - s]]]], {j, 1, Sqrt[n]}]];

%t T[n_] := CoefficientList[b[n], x];

%t T /@ Range[0, 14] // Flatten (* _Jean-François Alcover_, Feb 07 2021, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000007, A010052, A063725, A063691, A063730, A340481, A340905, A340906, A340915, A340946, A340947.

%Y Row sums give A006456.

%Y T(2n,n) gives A338464.

%Y Main diagonal gives A000012.

%Y Cf. A000290, A281704, A317665, A341040.

%K nonn,tabl

%O 0,18

%A _Alois P. Heinz_, Feb 03 2021