A176348 Triangle, read by rows: 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))).

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

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

Cf. A007318, A176298.

Sequence in context: A131778 A095713 A138072 * A176264 A195397 A173741

Adjacent sequences: A176345 A176346 A176347 * A176349 A176350 A176351

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 15 2010

Extensions

Edited by G. C. Greubel, Nov 23 2019

Status

approved