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A228708 Triangle T(n,k) read by rows: T(n,k) = number of permutations on 123...n with exactly one abc pattern and no aj pattern with j<=k, for n>=0, 0<=k<=n. 3
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 6, 6, 2, 0, 0, 27, 27, 12, 3, 0, 0, 110, 110, 55, 19, 4, 0, 0, 429, 429, 229, 91, 27, 5, 0, 0, 1638, 1638, 912, 393, 136, 36, 6, 0, 0, 6188, 6188, 3549, 1614, 612, 191, 46, 7, 0, 0, 23256, 23256, 13636, 6447, 2601, 897, 257, 57, 8, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

See Noonan-Zeilberger for precise definition.

LINKS

Table of n, a(n) for n=0..65.

J. Noonan and D. Zeilberger, [math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns. Also Adv. in Appl. Math. 17 (1996), no. 4, 381--407. MR1422065 (97j:05003).

FORMULA

T(n, k) = C(2n-k-1, n) - C(2n-k-1, n+3) + C(2n-2k-2, n-k-4) - C(2n-2k-2, n-k-1) + C(2n-2k-3, n-k-4) - C(2n-2k-3, n-k-2).

T(n, n-2) = n-2, T(n, k) = T(n, k+1) + T(n-1, k-1) + T(n-k, 2).

EXAMPLE

Triangle begins:

0

0,0

0,0,0

1,1,0,0

6,6,2,0,0

27,27,12,3,0,0

110,110,55,19,4,0,0

429,429,229,91,27,5,0,0

1638,1638,912,393,136,36,6,0,0

6188,6188,3549,1614,612,191,46,7,0,0

23256,23256,13636,6447,2601,897,257,57,8,0,0

...

PROG

(PARI) for(n=1, 15, for(k=1, n-2, print1(binomial(2*n-k-1, n)-binomial(2*n-k-1, n+3)+binomial(2*n-2*k-2, n-k-4)-binomial(2*n-2*k-2, n-k-1)+binomial(2*n-2*k-3, n-k-4)-binomial(2*n-2*k-3, n-k-2)", ")))

CROSSREFS

See A084249 for a curtailed version. See also A229158, A229160.

T(n, 1) = A003517(n+1). Cf. A001089.

Sequence in context: A122193 A098369 A078740 * A191504 A021155 A254245

Adjacent sequences:  A228705 A228706 A228707 * A228709 A228710 A228711

KEYWORD

nonn,tabl,easy

AUTHOR

N. J. A. Sloane, Sep 15 2013

STATUS

approved

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Last modified June 7 02:48 EDT 2020. Contains 334836 sequences. (Running on oeis4.)