2,6

T(n,k) = number of Dyck n-paths that start UU, contain no DUDU and no subpath of the form UUPDD with P a nonempty Dyck path and whose terminal descent has length n-k+2. For example, T(5,4)=2 counts UUDUUDUDDD, UUUDDUUDDD (each ending with exactly n-k+2=3 Ds). - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006

Martin Aigner, Motzkin numbers. European J. Combin. 19 (1998), 663-675.

J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No 62 (1978), 61-80.

R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.

Paul Peart and Wen-jin Woan, A divisibility property for a subgroup of Riordan matrices. Discrete Appl. Math. 98 (2000), 255-263.

a(n, k)=a(n, k-1)+a(n-1, k-1)+a(n-2, k-1), n>k >= 2.

1

0,1

0,1,2

0,0,2,4

0,0,1,5,9

0,0,0,3,12,21

0,0,0,1,9,30,51

0,0,0,0,4,25,76,127

0,0,0,0,1,14,69,196,323

a[2, 2]=1; a[n_, k_]/; Not[n>2 && 2<=k<=n] := 0; a[n_, k_]/; (n>2 && 2<=k<=n) := a[n, k] = a[n, k-1] + a[n-1, k-1] + a[n-2, k-1]; Table[a[n, k], {n, 2, 10}, {k, 2, n}] - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006

(PARI) T(n, k)=if(n==0&&k==0, 1, if(n<=0||k<=0||n<k, 0, T(n, k-1)+T(n-1, k-1)+T(n-2, k-1))) (from R. Stephan)

Main diagonal is A001006.

Other diagonals include A002026, A005322, A005323, A005324, A005325. Row sums are (essentially) A005043.

Sequence in context: A151669 A115509 A134312 this_sequence A135589 A158122 A028641

Adjacent sequences: A020471 A020472 A020473 this_sequence A020475 A020476 A020477

nonn,tabl,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 04 2000