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A304429 Triangle read by rows: T(n,k) (for 0 <= k <= floor(n/2)) is the number of permutations of length n that have k descents and avoid the patterns 321 and 2341. 0
1, 1, 1, 1, 1, 4, 1, 10, 2, 1, 20, 13, 1, 35, 49, 4, 1, 56, 140, 36, 1, 84, 336, 181, 8, 1, 120, 714, 670, 92, 1, 165, 1386, 2035, 578, 16, 1, 220, 2508, 5368, 2625, 224, 1, 286, 4290, 12727, 9633, 1688, 32, 1, 364, 7007, 27742, 30303, 9080, 528, 1, 455, 11011, 56485, 84721, 39041, 4640, 64 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Table of n, a(n) for n=0..63.

FORMULA

T(n, k) = Sum_{j=0..n-2k} (-1)^(n-j)*2^(3k-n+j-1)*binomial(3k+j, 3k)*binomial(k-1, n-2k-j) for k > 0.

G.f.: Sum_{n>=0} Sum_{k=0..floor(n/2)} T(n, k)*x^n*y^k = ((1-x)^2-x^2*y)/((1-x)^3-x^2(2-x)*y).

EXAMPLE

For n = 4 and k = 2, the T(4,2) = 2 permutations are 2143 and 3142.

Triangle T(n,k) begins:

  1;

  1;

  1,   1;

  1,   4;

  1,  10,    2;

  1,  20,   13;

  1,  35,   49,    4;

  1,  56,  140,   36;

  1,  84,  336,  181,   8;

  1, 120,  714,  670,  92;

  1, 165, 1386, 2035, 578, 16;

MATHEMATICA

T[n_, k_] :=

  If[k == 0, 1,

   Sum[(-1)^(n - j)*2^(3 k - n + j - 1)*Binomial[j + 3 k, 3 k]*

     Binomial[k - 1, n - 2 k - j], {j, 0, n - 2 k}]];

Flatten[Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 14}]]

CROSSREFS

Row sums give A001519.

Sequence in context: A138775 A209385 A121529 * A006370 A262370 A108759

Adjacent sequences:  A304426 A304427 A304428 * A304430 A304431 A304432

KEYWORD

nonn,tabf

AUTHOR

Colin Defant, May 12 2018

STATUS

approved

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Last modified July 22 00:47 EDT 2018. Contains 312889 sequences. (Running on oeis4.)