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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.
14
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, 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, 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
OFFSET
0,82
COMMENTS
Equals A181506 when the first column is removed. - Georg Fischer, Jul 26 2023
LINKS
FORMULA
T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000217(j)).
EXAMPLE
Triangle T(n,k) begins:
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, 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, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1;
MAPLE
h:= proc(n) option remember; `if`(n<1, 0,
`if`(issqr(8*n+1), n, h(n-1)))
end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);
MATHEMATICA
h[n_] := h[n] = If[n < 1, 0, If[IntegerQ @ Sqrt[8*n + 1], n, h[n - 1]]];
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[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, h[n]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000007, A010054 (for n>0), A052344, A063993, A319814, A319815, A319816, A319817, A319818, A319819, A319820.
Row sums give A007294.
T(2n,n) gives A319799.
Sequence in context: A079635 A037909 A181506 * A169987 A267611 A178666
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 28 2018
STATUS
approved