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A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A319797 #31 Jul 26 2023 21:16:11
%S A319797 1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,0,1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,1,
%T A319797 0,1,0,0,0,1,1,0,1,0,1,0,0,1,1,1,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,
%U A319797 1,0,1,1,1,1,0,1,0,1,0,0,1,2,1,1,1,1,1,0,1,0,1,0,0,1,1,2,1,1,1,1,1,0,1,0,1
%N A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A319797 Equals A181506 when the first column is removed. - _Georg Fischer_, Jul 26 2023
%H A319797 Alois P. Heinz, <a href="/A319797/b319797.txt">Rows n = 0..200, flattened</a>
%F A319797 T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000217(j)).
%e A319797 Triangle T(n,k) begins:
%e A319797   1;
%e A319797   0, 1;
%e A319797   0, 0, 1;
%e A319797   0, 1, 0, 1;
%e A319797   0, 0, 1, 0, 1;
%e A319797   0, 0, 0, 1, 0, 1;
%e A319797   0, 1, 1, 0, 1, 0, 1;
%e A319797   0, 0, 1, 1, 0, 1, 0, 1;
%e A319797   0, 0, 0, 1, 1, 0, 1, 0, 1;
%e A319797   0, 0, 1, 1, 1, 1, 0, 1, 0, 1;
%e A319797   0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
%e A319797   0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
%e A319797   0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1;
%p A319797 h:= proc(n) option remember; `if`(n<1, 0,
%p A319797       `if`(issqr(8*n+1), n, h(n-1)))
%p A319797     end:
%p A319797 b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
%p A319797       b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
%p A319797     end:
%p A319797 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
%p A319797 seq(T(n), n=0..20);
%t A319797 h[n_] := h[n] = If[n < 1, 0, If[IntegerQ @ Sqrt[8*n + 1], n, h[n - 1]]];
%t A319797 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[ x*b[n - i, h[Min[n - i, i]]]]];
%t A319797 T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, h[n]];
%t A319797 Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, May 27 2019, after _Alois P. Heinz_ *)
%Y A319797 Columns k=0-10 give: A000007, A010054 (for n>0), A052344, A063993, A319814, A319815, A319816, A319817, A319818, A319819, A319820.
%Y A319797 Row sums give A007294.
%Y A319797 T(2n,n) gives A319799.
%Y A319797 Cf. A000217, A117278, A181506, A243148, A319394.
%K A319797 nonn,tabl
%O A319797 0,82
%A A319797 _Alois P. Heinz_, Sep 28 2018