A172279 Triangle, read by rows: T(n, k) = ceiling( binomial(n, k)/(1 + (k - floor(n/2))^2) ).

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 6, 2, 1, 1, 3, 10, 5, 1, 1, 1, 2, 8, 20, 8, 2, 1, 1, 2, 11, 35, 18, 5, 1, 1, 1, 1, 6, 28, 70, 28, 6, 1, 1, 1, 1, 8, 42, 126, 63, 17, 4, 1, 1, 1, 1, 5, 24, 105, 252, 105, 24, 5, 1, 1, 1, 1, 6, 33, 165, 462, 231, 66, 17, 4, 1, 1, 1, 1, 4, 22, 99, 396, 924, 396, 99, 22, 4, 1, 1

Offset

0,5

Comments

Row sums are: {1, 2, 4, 7, 12, 21, 42, 74, 142, 264, 524, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n, k) = ceiling( binomial(n, k)/((1 + (k - floor(n/2))^2)).

T(2*n, n) = binomial(2*n,n) (cf. A000984). - Michel Marcus, May 22 2019

Example

Triangle begins as:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  2,  1;
  1, 2,  6,  2,   1;
  1, 3, 10,  5,   1,   1;
  1, 2,  8, 20,   8,   2,   1;
  1, 2, 11, 35,  18,   5,   1,  1;
  1, 1,  6, 28,  70,  28,   6,  1, 1;
  1, 1,  8, 42, 126,  63,  17,  4, 1, 1;
  1, 1,  5, 24, 105, 252, 105, 24, 5, 1, 1;

Mathematica

T[n_, k_]:= Ceiling[Binomial[n, k]/(1+(k-Floor[n/2])^2)];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(PARI)
{T(n, k) = ceil(binomial(n, k)/(1 + (k -floor(n/2))^2))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 21 2019
(Magma)
T:= func< n, k | Ceiling(Binomial(n, k)/(1 + (k - Floor(n/2))^2)) >;
[[T(n, k) : k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 21 2019
(Sage)
def T(n, k): return ceil(binomial(n, k)/(1 + (k -floor(n/2))^2))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 21 2019

Crossrefs

Cf. A000984 (central binomial coefficients).

Sequence in context: A006375 A327520 A184441 * A348285 A164953 A136622

Adjacent sequences: A172276 A172277 A172278 * A172280 A172281 A172282

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 30 2010

Extensions

Edited by G. C. Greubel, May 21 2019

Status

approved