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

0,2

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

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000984, A001850, A063007.

Sequence in context: A269336 A300700 A046521 * A060538 A260848 A110183

Adjacent sequences: A104681 A104682 A104683 * A104685 A104686 A104687

Keyword

nonn,tabl,easy

Author

Emeric Deutsch, Apr 24 2005

Status

approved