A306646 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. (k+1-x^k)/(1-x^k-x^(k+1)).

2, 3, 1, 4, 0, 3, 5, 0, 2, 4, 6, 0, 0, 3, 7, 7, 0, 0, 3, 2, 11, 8, 0, 0, 0, 4, 5, 18, 9, 0, 0, 0, 4, 0, 5, 29, 10, 0, 0, 0, 0, 5, 3, 7, 47, 11, 0, 0, 0, 0, 5, 0, 7, 10, 76, 12, 0, 0, 0, 0, 0, 6, 0, 4, 12, 123, 13, 0, 0, 0, 0, 0, 6, 0, 4, 3, 17, 199

Offset: 0

Author

Seiichi Manyama, Mar 03 2019

Keywords

• nonn
• tabl

Examples

			A(6,1) = 6*Sum_{j=1..6} binomial(j,6-j)/j = 6*(1/3+3/2+1+1/6) = 18.
A(6,2) = 6*Sum_{j=1..3} binomial(j,6-2*j)/j = 6*(1/2+1/3) = 5.
Square array begins:
    2,  3, 4, 5, 6, 7, 8, 9, 10, 11, ...
    1,  0, 0, 0, 0, 0, 0, 0,  0,  0, ...
    3,  2, 0, 0, 0, 0, 0, 0,  0,  0, ...
    4,  3, 3, 0, 0, 0, 0, 0,  0,  0, ...
    7,  2, 4, 4, 0, 0, 0, 0,  0,  0, ...
   11,  5, 0, 5, 5, 0, 0, 0,  0,  0, ...
   18,  5, 3, 0, 6, 6, 0, 0,  0,  0, ...
   29,  7, 7, 0, 0, 7, 7, 0,  0,  0, ...
   47, 10, 4, 4, 0, 0, 8, 8,  0,  0, ...
   76, 12, 3, 9, 0, 0, 0, 9,  9,  0, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns 1-9 give A000032, A001608, A050443, A087935, A087936, A306755, A306756, A306757, A306758.

Cf. A306713, A306735.

Programs

Mathematica

T[0, k_] := k + 1; T[n_, k_] := n *Sum[Binomial[j, n - k*j]/j, {j, 1, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

Formula

A(0,k) = k+1 and A(n,k) = n*Sum_{j=1..floor(n/k)} binomial(j,n-k*j)/j for n > 0.

A(n,k) = (k+1)*A306713(n,k) - A306713(n-k,k) for n >= k.