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A068218 Triangle of numbers of square lattice walks that start and end at origin after 2k steps and contain exactly r steps to the east, not touching origin at intermediate stages. 0
1, 2, 2, 2, 16, 2, 4, 84, 84, 4, 10, 400, 1056, 400, 10, 28, 1820, 9184, 9184, 1820, 28, 84, 8064, 66276, 126720, 66276, 8064, 84, 264, 35112, 426888, 1329768, 1329768, 426888, 35112, 264, 858, 151008, 2546544, 11737440, 19123776, 11737440 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The given recurrences do not provide a means to calculate T(2r,r). But T(2r,r) is computable by the formula relating T(k,r) to A069466(k,r).

LINKS

Table of n, a(n) for n=0..41.

FORMULA

T(k, r) = 2*(2k-3)/(k-2r) * ( T(k-1, r) - T(k-1, r-1) ), for k > 2r. T(1, 0)=2, T(1, 1)=2 Sum[T(k, r), r=0, ..., k] = A054474(k) T(k, r)=A069466(k, r) - Sum[ Sum[ T(i, j)*A069466(k-i, r-j), j=0...r], i=1, k-1]

EXAMPLE

T(3,1)=84 because there are 84 distinct lattice walks of length 2*3=6 starting and ending at the origin and containing exactly 1 step to the east and not touching origin at intermediate steps. Let E, W, S, N denote the 4 possible directions, then NNEWSS and NWSSNE are examples of such walks.

MATHEMATICA

A069466[k_, r_] := Binomial[2 k, k]*Binomial[k, r]^2; t[k_, r_] := t[k, r] = A069466[k, r] - Sum[Sum[t[i, j]*A069466[k - i, r - j], {j, 0, r}], {i, 1, k - 1}]; Table[t[k, r], {k, 0, 8}, {r, 0, k}] // Flatten (* Jean-Fran├žois Alcover, Nov 21 2012, from formula *)

CROSSREFS

T(k, 0) = A002420(k) = A069466(k)/(2k-1).

Cf. A054474 (row sums).

Sequence in context: A129409 A025521 A305109 * A098919 A161748 A195871

Adjacent sequences:  A068215 A068216 A068217 * A068219 A068220 A068221

KEYWORD

easy,nice,nonn,tabl

AUTHOR

Martin Wohlgemuth (mail(AT)matroid.com), Mar 24 2002

STATUS

approved

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Last modified October 18 07:42 EDT 2019. Contains 328146 sequences. (Running on oeis4.)