A306444 A(n,k) = binomial((2*k+1)*n+2, k*n+1)/((2*k+1)*n+2), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

1, 1, 1, 1, 2, 1, 1, 5, 7, 1, 1, 14, 66, 30, 1, 1, 42, 715, 1144, 143, 1, 1, 132, 8398, 49742, 22610, 728, 1, 1, 429, 104006, 2340135, 3991995, 482885, 3876, 1, 1, 1430, 1337220, 115997970, 757398510, 347993910, 10855425, 21318, 1

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..81, flattened

Example

Square array begins:
   1,    1,        1,           1,              1, ...
   1,    2,        5,          14,             42, ...
   1,    7,       66,         715,           8398, ...
   1,   30,     1144,       49742,        2340135, ...
   1,  143,    22610,     3991995,      757398510, ...
   1,  728,   482885,   347993910,   267058714626, ...
   1, 3876, 10855425, 32018897274, 99543581789652, ...

Mathematica

A[n_, k_]:= Binomial[(2*k+1)*n+2, k*n+1]/((2*k+1)*n+2); Table[A[k, n-k], {n, 0, 12}, {k, 0, n}] (* G. C. Greubel, Feb 16 2019 *)

Prog

(PARI) {A(n, k) = binomial((2*k+1)*n+2, k*n+1)/((2*k+1)*n+2)};
for(n=0, 12, for(k=0, n, print1(A(k, n-k), ", "))) \\ G. C. Greubel, Feb 16 2019
(Magma) [[Binomial((2*(n-k)+1)*k+2, k*(n-k)+1)/((2*(n-k)+1)*k+2): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 16 2019
(Sage) [[binomial((2*(n-k)+1)*k+2, k*(n-k)+1)/((2*(n-k)+1)*k+2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Feb 16 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial((2*(n-k)+1)*k+2, k*(n-k)+1)/((2*(n-k)+1)*k+2) ))); # G. C. Greubel, Feb 16 2019

Crossrefs

Columns 0-1 give A000012, A006013.

Rows 0-5 give A000012, A000108(n+1), A065097(n+1), A265101, A265102, A265103.

Sequence in context: A064814 A051012 A064644 * A090210 A248925 A168131

Adjacent sequences: A306441 A306442 A306443 * A306445 A306446 A306447

Keyword

nonn,tabl

Author

Seiichi Manyama, Feb 15 2019

Status

approved