A341040 - OEIS

%I #40 Feb 15 2021 07:27:11

%S 1,0,1,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,1,0,

%T 0,0,0,1,0,0,0,1,0,0,0,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0,0,1,1,0,0,0,0,0,

%U 1,0,0,0,1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,0,1,0,0,1,1,0,0,0,1,0,0,0,0,1,1

%N Number T(n,k) of partitions of n into k distinct nonzero squares; triangle T(n,k), n>=0, 0<=k<=A248509(n), read by rows.

%C T(n,k) is defined for n, k >= 0.  The triangle contains only the terms with 0 <= k <= A248509(n).  T(n,k) = 0 for k > A248509(n).

%H Alois P. Heinz, <a href="/A341040/b341040.txt">Rows n = 0..4000, flattened</a>

%F T(n,k) = [x^n*y^k] Product_{j>=1} (1 + y*x^(j^2)).

%F T(A000330(n),n) = 1.

%F Row n = [0] <=> n in { A001422 }.

%F Sum_{k>=0} 2^k * T(n,k) = A279360(n).

%F Sum_{k>=0} k * T(n,k) = A281542(n).

%F Sum_{k>=0} (-1)^k * T(n,k) = A276516(n).

%e T(62,3) = 2 is the first term > 1 and counts partitions [49,9,4] and [36,25,1].

%e Triangle T(n,k) begins:

%e   1;

%e   0, 1;

%e   0;

%e   0;

%e   0, 1;

%e   0, 0, 1;

%e   0;

%e   0;

%e   0;

%e   0, 1;

%e   0, 0, 1;

%e   0;

%e   0;

%e   0, 0, 1;

%e   0, 0, 0, 1;

%e   ...

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       b(n, i-1)+`if`(i^2>n, 0, expand(b(n-i^2, i-1)*x))))

%p     end:

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

%p seq(T(n), n=0..45);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,

%t b[n, i - 1] + If[i^2 > n, 0, Expand[b[n - i^2, i - 1]*x]]]];

%t T[n_] := CoefficientList[b[n, Floor@Sqrt[n]], x] /. {} -> {0};

%t T /@ Range[0, 45] // Flatten (* _Jean-François Alcover_, Feb 15 2021, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000007, A010052 (for n>0), A025441, A025442, A025443, A025444, A340988, A340998, A340999, A341000, A341001.

%Y Row sums give A033461.

%Y Cf. A000290, A000330, A001422, A243148, A248509, A276516, A279360, A281542, A337165.

%K nonn,look,tabf

%O 0

%A _Alois P. Heinz_, Feb 03 2021