A330965 Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 10, 20, 14, 1, 5, 14, 35, 70, 42, 1, 6, 18, 50, 126, 252, 132, 1, 7, 22, 65, 182, 462, 924, 429, 1, 8, 26, 80, 238, 672, 1716, 3432, 1430, 1, 9, 30, 95, 294, 882, 2508, 6435, 12870, 4862, 1, 10, 34, 110, 350, 1092, 3300, 9438, 24310, 48620, 16796

Offset

0,5

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Formula

A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).

A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.

G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).

Example

Array begins:
====================================================
n\k |   0    1    2    3     4     5     6     7
----+-----------------------------------------------
  0 |   1    1    1    1     1     1     1     1 ...
  1 |   1    2    3    4     5     6     7     8 ...
  2 |   2    6   10   14    18    22    26    30 ...
  3 |   5   20   35   50    65    80    95   110 ...
  4 |  14   70  126  182   238   294   350   406 ...
  5 |  42  252  462  672   882  1092  1302  1512 ...
  6 | 132  924 1716 2508  3300  4092  4884  5676 ...
  7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
  ...

Prog

(PARI) T(n, k)={(1 + k*n)*binomial(2*n, n)/(n+1)}

Crossrefs

Columns k=0..12 are A000108, A000984, A001700, A051924, A051944, A051945, A050476, A050477, A050478, A050479, A050489, A050490, A050491.

Sequence in context: A340106 A259824 A065173 * A098474 A153199 A056860

Adjacent sequences: A330962 A330963 A330964 * A330966 A330967 A330968

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 04 2020

Status

approved