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A084249 Triangle T(n,k) read by rows: permutations on 123...n with one abc pattern and no aj pattern with j<=k, n>2, k<n-1. 3
1, 6, 2, 27, 12, 3, 110, 55, 19, 4, 429, 229, 91, 27, 5, 1638, 912, 393, 136, 36, 6, 6188, 3549, 1614, 612, 191, 46, 7, 23256, 13636, 6447, 2601, 897, 257, 57, 8, 87210, 52020, 25332, 10695, 3951, 1260, 335, 69, 9, 326876, 197676, 98532 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

See A228708 for further information.

LINKS

Table of n, a(n) for n=3..50.

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

Full 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 A228708 for the full triangle.

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

Sequence in context: A036173 A142707 A176965 * A176591 A191703 A096039

Adjacent sequences:  A084246 A084247 A084248 * A084250 A084251 A084252

KEYWORD

nonn,tabl,easy

AUTHOR

Ralf Stephan, May 21 2003

STATUS

approved

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Last modified February 22 11:50 EST 2018. Contains 299452 sequences. (Running on oeis4.)