A241262 Array t(n,k) = binomial(n*k, n+1)/n, where n >= 1 and k >= 2, read by ascending antidiagonals.

1, 2, 3, 5, 10, 6, 14, 42, 28, 10, 42, 198, 165, 60, 15, 132, 1001, 1092, 455, 110, 21, 429, 5304, 7752, 3876, 1020, 182, 28, 1430, 29070, 57684, 35420, 10626, 1995, 280, 36, 4862, 163438, 444015, 339300, 118755, 24570, 3542, 408, 45, 16796, 937365, 3506100, 3362260, 1391280, 324632, 50344, 5850, 570, 55

Offset

1,2

Comments

About the "root estimation" question asked in MathOverflow, one can check (at least numerically) that, for instance with k = 4 and a = 1/11, the series a^-1 + (k - 1) + Sum_{n>=} (-1)^n*binomial(n*k, n+1)/n*a^n evaluates to the positive solution of x^k = (x+1)^(k-1).

Row 1 is A000217 (triangular numbers),

Row 2 is A006331 (twice the square pyramidal numbers),

Row 3 is A067047(3n) = lcm(3n, 3n+1, 3n+2, 3n+3)/12 (from column r=4 of A067049),

Row 4 is A222715(2n) = (n-1)*n*(2n-1)*(4n-3)*(4n-1)/15,

Row 5 is not in the OEIS.

Column 1 is A000108 (Catalan numbers),

Column 2 is A007226 left shifted 1 place,

Column 4 is A007228 left shifted 1 place,

Column 5 is A124724 left shifted 1 place,

Column 6 is not in the OEIS.

References

N. S. S. Gu, H. Prodinger, S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat. 31 (2010) 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2

Links

Table of n, a(n) for n=1..55.

MathOverflow, Root estimation

Example

Array begins:
    1,    3,     6,     10,      15,      21, ...
    2,   10,    28,     60,     110,     182, ...
    5,   42,   165,    455,    1020,    1995, ...
   14,  198,  1092,   3876,   10626,   24570, ...
   42, 1001,  7752,  35420,  118755,  324632, ...
  132, 5304, 57684, 339300, 1391280, 4496388, ...
  etc.

Mathematica

t[n_, k_] := Binomial[n*k, n+1]/n; Table[t[n-k+2, k], {n, 1, 10}, {k, 2, n+1}] // Flatten

Crossrefs

Cf. A000217, A006331, A000108, A007226, A007228, A067047, A067049, A124724, A222715.

Sequence in context: A250745 A109204 A250747 * A244489 A367297 A286144

Adjacent sequences: A241259 A241260 A241261 * A241263 A241264 A241265

Keyword

nonn,tabl,easy

Author

Jean-François Alcover, Apr 18 2014

Status

approved