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A236406 Triangle read by rows: number of (1-2-3)-avoiding permutations on n letters with k peaks. 2
1, 1, 2, 3, 2, 4, 10, 5, 32, 5, 6, 84, 42, 7, 198, 210, 14, 8, 438, 816, 168, 9, 932, 2727, 1152, 42, 10, 1936, 8250, 5940, 660, 11, 3962, 23276, 25630, 5775, 132, 12, 8034, 62400, 97812, 37180, 2574, 13, 16200, 160953, 341224, 196625, 27456, 429, 14, 32556, 402906, 1111656, 905086, 212212, 10010 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is a convolution of A091156 with itself (see the Pudwell link below).

LINKS

Alois P. Heinz, Rows n = 0..120, flattened

A. M. Baxter, Refining enumeration schemes to count according to permutation statistics, arXiv preprint arXiv:1401.0337 [math.CO], 2014.

M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.

L. Pudwell, On the distribution of peaks (and other statistics), 2018.

FORMULA

T(2*n+2,n) = A276666(n+2) = (n+1)*A000108(n+2). - Alois P. Heinz, Apr 27 2018

G.f.: G(q,z) = - (-2z^3q^2+4z^3q-2z^3-2z^2q+2z^2-1+sqrt(-4z^2q+4z^2-4z+1))/(2z(zq-z+1)^2). (See the Pudwell link above.)

EXAMPLE

Triangle begins:

   1;

   1;

   2;

   3,    2;

   4,   10;

   5,   32,    5;

   6,   84,   42;

   7,  198,  210,   14;

   8,  438,  816,  168;

   9,  932, 2727, 1152,  42;

  10, 1936, 8250, 5940, 660;

  ...

MATHEMATICA

m = maxExponent = 15;

G = -(-2 z^3 q^2 + 4z^3 q - 2z^3 - 2z^2 q + 2z^2 - 1 + Sqrt[-4z^2 q + 4z^2 - 4z + 1])/(2z (z q - z + 1)^2);

CoefficientList[# + O[q]^m, q]& /@ CoefficientList[G + O[z]^m, z]// Flatten (* Jean-Fran├žois Alcover, Aug 06 2018 *)

CROSSREFS

Row sums give A000108.

Cf. A091894, A276666.

Sequence in context: A034800 A082771 A127157 * A247497 A202714 A022662

Adjacent sequences:  A236403 A236404 A236405 * A236407 A236408 A236409

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Jan 31 2014

EXTENSIONS

More terms from Alois P. Heinz, Apr 26 2018

STATUS

approved

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Last modified April 21 04:56 EDT 2019. Contains 322310 sequences. (Running on oeis4.)