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A110537
Symmetric number square associated to ceiling(k^n/n^k), read by antidiagonals.
4
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.
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
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Jul 25 2005
STATUS
approved