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Search: id:A104684
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| A104684 |
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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. |
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+0
7
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| 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
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row sums are the central Delannoy numbers (A001850). T(n,0)=A000984(n) (the central binomial numbers). Alternating row sums = 1 (Math. Magazine, 77, No. 1, 2004, p. 321). Mirror image 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 DELEHAM (kolotoko(AT)wanadoo.fr), Apr 25 2005
T(2n,n)=C(2n,n)C(3n,n)=C(n,n)C(2n,n)C(3n,n)=A006480(n). - Paul Barry (pbarry(AT)wit.ie), Mar 14 2006
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). [From Paul Barry (pbarry(AT)wit.ie), Apr 19 2009]
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REFERENCES
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R. A. Sulanke, Objects counted by the central Delannoy numbers. J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
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LINKS
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T. D. Noe, Rows n=0..100 of triangle, flattened
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FORMULA
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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 (pbarry(AT)wit.ie), Mar 14 2006
G.f.: 1/(1-xy-2x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x... (continued fraction); [From Paul Barry (pbarry(AT)wit.ie), Jan 06 2009]
T(n,k)=sum{j=0..n, C(n,j)^2*C(j,k)}. [From Paul Barry (pbarry(AT)wit.ie), May 28 2009]
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EXAMPLE
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T(2,1)=6 because we have NED, NDE, EDN, END, DEN and DNE.
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MAPLE
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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
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CROSSREFS
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Cf. A001850, A063007, A000984.
Cf. A063007
Sequence in context: A052296 A019538 A046521 this_sequence A060538 A110183 A110098
Adjacent sequences: A104681 A104682 A104683 this_sequence A104685 A104686 A104687
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 24 2005
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