A104856 Triangle read by rows: T(n,k) = binomial(n,k)*binomial(k,floor(k/2))*binomial(n-k,floor((n-k)/2)) (0<=k<=n).

1, 1, 1, 2, 2, 2, 3, 6, 6, 3, 6, 12, 24, 12, 6, 10, 30, 60, 60, 30, 10, 20, 60, 180, 180, 180, 60, 20, 35, 140, 420, 630, 630, 420, 140, 35, 70, 280, 1120, 1680, 2520, 1680, 1120, 280, 70, 126, 630, 2520, 5040, 7560, 7560, 5040, 2520, 630, 126, 252, 1260, 6300

Offset: 0

Author

Emeric Deutsch, Apr 23 2005

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of paths in the first quadrant, starting from the origin, with unit steps up, down, right, or left, having a total of n steps, exactly k of which are vertical (up or down). Example: T(3,2)=6 because we have NNE, NEN, ENN, NSE, ENS and NES. [Emeric Deutsch, Nov 22 2008]

Links

• David M. Bloom et al., A Convolution of Middle Binomial Coefficients: Problem 10921, Amer. Math. Monthly 110, (2003), 958-959.
• E. Deutsch and D. Lovit, Problem 1739, Math. Magazine, vol. 80, No. 1, 2007, p. 80. [_Emeric Deutsch_, Nov 22 2008]

Crossrefs

Row sums yield A005566. T(n, 0)=T(n, n)=A001405(n).

Cf. A005566, A001405.

Programs

Maple

T:=(n,k)->binomial(n,k)*binomial(k,floor(k/2))*binomial(n-k,floor((n-k)/2)): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Formula

T(n, k) = binomial(n, k)*binomial(k, floor(k/2))*binomial(n-k, floor((n-k)/2)) (0<=k<=n).