A110537 Symmetric number square associated to ceiling(k^n/n^k), read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 7, 5, 3, 1, 1, 4, 7, 9, 9, 7, 4, 1, 1, 5, 11, 15, 14, 15, 11, 5, 1, 1, 8, 18, 25, 24, 24, 25, 18, 8, 1, 1, 12, 35, 47, 40, 47, 40, 47, 35, 12, 1, 1, 18, 72, 102, 79, 81, 81, 79, 102, 72, 18, 1, 1, 30, 152, 237, 183, 168

Offset

1,5

Comments

Row sums of triangle are A110538. Diagonal sums are A110539. The row sums of the inverse of the triangle may be A000007.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

Number square T(n, k) = Sum_{j=1..min(n, k)} ceiling(j^n/n^j)*ceiling(j^k/k^j).

As a number triangle, T(n, k) = if(k<=n, Sum_{j=1..min(n-k+1, k)} ceiling(j^(n-k+1)/(n-k+1)^j)*ceiling(j^k/k^j), 0).

Example

As a number square, rows begin
1,1,1,1,1,1,1,...
1,2,2,2,3,3,4,...
1,2,3,4,5,7,11,...
1,2,4,7,9,15,25,...
1,3,5,9,14,24,40,...
1,3,7,15,24,47,81,...
As a number triangle, rows begin
1;
1,1;
1,2,1;
1,2,2,1;
1,2,3,2,1;
1,3,4,4,3,1;
1,3,5,7,5,3,1;

Mathematica

T[n_, k_] := If[k <= n, Sum[Ceiling[j^(n - k + 1)/(n - k + 1)^j]*Ceiling[j^k/k^j], {j, 1, Min[n - k + 1, k]}], 0]; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Aug 30 2017 *)

Prog

(PARI) for(n=1, 20, for(k=1, n, print1(if(k<=n, sum(j=1, min(n-k+1, k), ceil(j^(n-k+1)/(n-k+1)^j)*ceil(j^k/k^j)), 0), ", "))) \\ G. C. Greubel, Aug 30 2017

Crossrefs

Sequence in context: A204026 A300119 A323211 * A144434 A322057 A323767

Adjacent sequences: A110534 A110535 A110536 * A110538 A110539 A110540

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 25 2005

Status

approved