OFFSET
0,5
COMMENTS
Row sums are: {1, 2, 8, 14, 56, 82, 298, 520, 1408, 2594, 7484, ...}.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n, k) = binomial(n, k)*(1 +2*(n+1) -(k+1)*floor((n+1)/(k+1)) -(n-k+1)* floor((n+1)/(n-k+1))).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 6, 6, 1;
1, 12, 30, 12, 1;
1, 10, 30, 30, 10, 1;
1, 18, 60, 140, 60, 18, 1;
1, 14, 105, 140, 140, 105, 14, 1;
1, 24, 84, 280, 630, 280, 84, 24, 1;
1, 18, 144, 504, 630, 630, 504, 144, 18, 1;
1, 30, 225, 840, 1260, 2772, 1260, 840, 225, 30, 1;
MAPLE
T:=binomial(n, k)*(2*n+3 -(k+1)*floor((n+1)/(k+1)) -(n-k+1)* floor((n+1)/(n-k+1))); seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 23 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= Binomial[n, k]*(2*n+3 -(k+1)*Floor[(n+1)/(k+1)] -(n - k+1)*Floor[(n+1)/(n-k+1)]); Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(PARI) T(n, k) = binomial(n, k)*(2*n+3 -(k+1)*((n+1)\(k+1)) -(n-k+1)* ((n+1)\(n-k+1))); \\ G. C. Greubel, Nov 23 2019
(Magma) [Binomial(n, k)*(2*n+3 -(k+1)*Floor((n+1)/(k+1)) -(n-k+1)* Floor((n+1)/(n-k+1))): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 23 2019
(Sage) [[binomial(n, k)*(2*n+3 -(k+1)*floor((n+1)/(k+1)) -(n-k+1)* floor((n+1)/(n-k+1))) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 23 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*(2*n+3 -(k+1)*Int((n+1)/(k+1)) -(n-k+1)*Int((n+1)/(n-k+1))) ))); # G. C. Greubel, Nov 23 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 15 2010
EXTENSIONS
Edited by G. C. Greubel, Nov 23 2019
STATUS
approved