0,4

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.

Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Triangle begins:

1;

1, 1;

2, 2, 1;

4, 6, 3, 1;

10, 15, 12, 4, 1;

25, 45, 36, 20, 5, 1;

70, 126, 126, 70, 30, 6, 1;

...

c[n_, k_] := Boole[k <= n] Binomial[2n - k, n] (k + 1)/(n + 1);

T[n_, k_] := Module[{nn, kk}, If[OddQ[n], nn = (n + 1)/2, nn = n/2]; If[OddQ[k], kk = (k - 1)/2, kk = k/2]; If [OddQ[n], If[OddQ[k], c[nn + kk, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk, 2kk + 1]], If[OddQ[k], c[nn + kk + 1, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk + 1, 2kk + 1]]]];

Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 04 2018, from PARI *)

(PARI) C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);

T(n, k) = {my(nn, kk); if (n % 2, nn = (n+1)/2, nn = n/2); if (k % 2, kk = (k-1)/2, kk = k/2); if ((n % 2), if (k % 2, C(nn+kk, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk, 2*kk+1)), if (k % 2, C(nn+kk+1, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk+1, 2*kk+1))); } \\ Michel Marcus, Feb 13 2014

Cf. A033184, A228334, A228335, A228337.

Sequence in context: A175136 A091869 A112307 * A111062 A369632 A363493

Adjacent sequences: A228333 A228334 A228335 * A228337 A228338 A228339

nonn,tabl

N. J. A. Sloane, Aug 26 2013

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

approved