|
%I #53 Apr 19 2022 07:36:56
%S 1,2,1,6,6,1,20,30,12,1,70,140,90,20,1,252,630,560,210,30,1,924,2772,
%T 3150,1680,420,42,1,3432,12012,16632,11550,4200,756,56,1,12870,51480,
%U 84084,72072,34650,9240,1260,72,1,48620,218790,411840,420420,252252
%N 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.
%C 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.
%C Row reversed version of A063007.
%C 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
%C Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with decreasing powers of x.
%C Coefficient array of x^n*Legendre_P(n,2/x+1). - _Paul Barry_, Apr 19 2009
%H T. D. Noe, <a href="/A104684/b104684.txt">Rows n=0..100 of triangle, flattened</a>
%H Michel Bataille, <a href="http://www.jstor.org/stable/3219295?seq=2#page_scan_tab_contents">Quickie Q.944</a>, Maths. Magazine, 77, No. 4, p. 321, <a href="http://www.jstor.org/stable/3219295?seq=8#page_scan_tab_contents">Answer A.944</a>, Maths. Magazine, 77, No. 4, p. 327.
%H H. J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>.
%H Ömür Deveci and Anthony G. Shannon, <a href="https://www.montis.pmf.ac.me/allissues/50/Mathematica-Montisnigri-L-4.pdf">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri, Volume L, 2021.
%H Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
%F T(n, k) = binomial(n, k)*binomial(2n-k, n) (0 <= k <= n).
%F G.f.: G(t, z) = 1/sqrt((1-tz)^2 - 4z).
%F T(n,k) = binomial(2(n-k),n-k)*binomial(2n-k,k). - _Paul Barry_, Mar 14 2006
%F 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
%F 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
%F T(n,k) = Sum_{j=0..n} C(n,j)^2*C(j,k). - _Paul Barry_, May 28 2009
%F T(n,k) = [x^k]F(-n,-n;1;1+x). - _Paul Barry_, Oct 05 2010
%F T(n,k) = (n-k+1)*A060693(n,k). - _Peter Luschny_, May 17 2011
%F T(n,k) = A054142(n,k)*A000984(n-k). - _Philippe Deléham_, Nov 19 2011.
%F T(n,k) = abs(A130595(n,k)*A092392(n,k)). - _Reinhard Zumkeller_, Dec 20 2013
%e T(2,1)=6 because we have NED, NDE, EDN, END, DEN and DNE.
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 ...
%e 0: 1
%e 1: 2 1
%e 2: 6 6 1
%e 3: 20 30 12 1
%e 4: 70 140 90 20 1
%e 5: 252 630 560 210 30 1
%e 6: 924 2772 3150 1680 420 42 1
%e 7: 3432 12012 16632 11550 4200 756 56 1
%e 8: 12870 51480 84084 72072 34650 9240 1260 72 1
%e ...
%e row n=9: 48620 218790 411840 420420 252252 90090 18480 1980 90 1,
%e row n=10: 184756 923780 1969110 2333760 1681680 756756 210210 34320 2970 110 1.
%e ... reformatted by _Wolfdieter Lang_, Sep 11 2016
%p 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
%t T[n_, k_] := Binomial[n, k] Binomial[2n-k, n];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 19 2018 *)
%o (Haskell)
%o a104684 n k = a104684_tabl !! n !! k
%o a104684_row n = a104684_tabl !! n
%o a104684_tabl = map (map abs) $
%o zipWith (zipWith (*)) a130595_tabl a092392_tabl
%o -- _Reinhard Zumkeller_, Dec 20 2013
%Y Cf. A000984, A001850, A063007.
%K nonn,tabl,easy
%O 0,2
%A _Emeric Deutsch_, Apr 24 2005
|
