A104684 Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.

1, 2, 1, 6, 6, 1, 20, 30, 12, 1, 70, 140, 90, 20, 1, 252, 630, 560, 210, 30, 1, 924, 2772, 3150, 1680, 420, 42, 1, 3432, 12012, 16632, 11550, 4200, 756, 56, 1, 12870, 51480, 84084, 72072, 34650, 9240, 1260, 72, 1, 48620, 218790, 411840, 420420, 252252

Offset: 0

Author

Emeric Deutsch, Apr 24 2005

Keywords

• nonn
• tabl
• easy

Comments

Row sums are the central Delannoy numbers (A001850). T(n,0)=A000984(n) (the central binomial numbers). Alternating row sums = 1 See the Bataille link.

Row reversed version of A063007.

Another version of [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] = 1; 0, 1; 0, 2, 1; 0, 6, 6, 1; 0, 20, 30, 12, 1; 0, 70, 140, 90, 20, 1; ..., where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 25 2005

Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with decreasing powers of x.

Coefficient array of x^n*Legendre_P(n,2/x+1). - Paul Barry, Apr 19 2009

Examples

			T(2,1)=6 because we have NED, NDE, EDN, END, DEN and DNE.
The triangle T(n, k) begins:
n\k    0     1     2     3     4    5    6  7 8 ...
0:     1
1:     2     1
2:     6     6     1
3:    20    30    12     1
4:    70   140    90    20    1
5:   252   630   560   210   30     1
6:   924  2772  3150  1680  420    42    1
7:  3432 12012 16632 11550 4200   756   56  1
8: 12870 51480 84084 72072 34650 9240 1260 72 1
...
row n=9: 48620 218790 411840 420420 252252 90090 18480 1980 90 1,
row n=10: 184756 923780 1969110 2333760 1681680 756756 210210 34320 2970 110 1.
... reformatted by _Wolfdieter Lang_, Sep 11 2016
		

Links

• T. D. Noe, Rows n=0..100 of triangle, flattened
• Michel Bataille, Quickie Q.944, Maths. Magazine, 77, No. 4, p. 321, Answer A.944, Maths. Magazine, 77, No. 4, p. 327.
• H. J. Brothers, Pascal's Prism: Supplementary Material.
• Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri, Volume L, 2021.
• Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Crossrefs

Cf. A000984, A001850, A063007.

Programs

Haskell

a104684 n k = a104684_tabl !! n !! k
a104684_row n = a104684_tabl !! n
a104684_tabl = map (map abs) $
               zipWith (zipWith (*)) a130595_tabl a092392_tabl
-- Reinhard Zumkeller, Dec 20 2013

Maple

T:=(n,k)->binomial(n,k)*binomial(2*n-k,n): for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

T[n_, k_] := Binomial[n, k] Binomial[2n-k, n];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

Formula

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

G.f.: G(t, z) = 1/sqrt((1-tz)^2 - 4z).

T(n,k) = binomial(2(n-k),n-k)*binomial(2n-k,k). - Paul Barry, Mar 14 2006

T(2n,n) = C(2n,n)*C(3n,n) = C(n,n)*C(2n,n)*C(3n,n) = A006480(n). - Paul Barry, Mar 14 2006

G.f.: 1/(1-xy-2x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x... (continued fraction). - Paul Barry, Jan 06 2009

T(n,k) = Sum_{j=0..n} C(n,j)^2*C(j,k). - Paul Barry, May 28 2009

T(n,k) = [x^k]F(-n,-n;1;1+x). - Paul Barry, Oct 05 2010

T(n,k) = (n-k+1)*A060693(n,k). - Peter Luschny, May 17 2011

T(n,k) = A054142(n,k)*A000984(n-k). - Philippe Deléham, Nov 19 2011.

T(n,k) = abs(A130595(n,k)*A092392(n,k)). - Reinhard Zumkeller, Dec 20 2013