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A110098 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n, having k return steps to the line y=x from the line y=x+1 (i.e. E steps from the line y=x+1 to the line y=x; a Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1)). 2
1, 2, 1, 6, 6, 1, 22, 30, 10, 1, 90, 146, 70, 14, 1, 394, 714, 430, 126, 18, 1, 1806, 3534, 2490, 938, 198, 22, 1, 8558, 17718, 14002, 6314, 1734, 286, 26, 1, 41586, 89898, 77550, 40054, 13338, 2882, 390, 30, 1, 206098, 461010, 426150, 244790, 94554 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The row sums are the central Delannoy numbers (A001850).

Column 0 yields the large Schroeder numbers (A006318).

Column 1 yields A006320.

Column k has g.f. z^k*R^(2*k+1), where R = 1+z*R+z*R^2 is the g.f. of the large Schroeder numbers (A006318).

REFERENCES

R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.

LINKS

Table of n, a(n) for n=0..49.

FORMULA

T(n, k) = ((2*k+1)/(n-k))*sum(binomial(n-k, j)*binomial(n+k+j, n-k-1), j=0..n-k) for k < n; T(n, n) = 1; T(n, k) = 0 for k > n. G.f. = R/(1-t*z*R^2), where R = 1+z*R+z*R^2 is the g.f. of the large Schroeder numbers (A006318).

sum(k*T(n, k), k=0..n) = A110099(n).

T(n, k) = A033877(n-k+1, n+k+1). - Johannes W. Meijer, Sep 05 2013

EXAMPLE

T(2, 1) = 6 because we have DN(E), N(E)D, N(E)EN, ND(E), NNE(E) and ENN(E) (the return E steps are shown between parentheses).

Triangle begins:

1;

2,1;

6,6,1;

22,30,10,1;

90,146,70,14,1;

MAPLE

T := proc(n, k) if k=n then 1 else ((2*k+1)/(n-k))*sum(binomial(n-k, j)*binomial(n+k+j, n-k-1), j=0..n-k) fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

CROSSREFS

Cf. A001850, A006318, A006320, A110099, A110107, A033877.

Sequence in context: A104684 A060538 A110183 * A244888 A130561 A157400

Adjacent sequences:  A110095 A110096 A110097 * A110099 A110100 A110101

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jul 11 2005

STATUS

approved

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Last modified December 22 22:37 EST 2014. Contains 252372 sequences.