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A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks. 6
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 9, 9, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 4, 16, 24, 36, 24, 16, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 6, 36, 90, 225 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Rows 2,4,6,... give A088459.

Diagonal sums are in A088518. - Philippe Deléham, Jan 04 2009

Row sums are in A001405. - Philippe Deléham, Jan 04 2009

Subtriangle (1 <= k <= n) of triangle T(n,k), 0 <= k <= n, read by rows, given by A101455 DELTA A056594 := [0,1,0,-1,0,1,0,-1,0,1,0,-1,0,...] DELTA [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009

Also, number of symmetric noncrossing partitions of an n-set with k blocks. - Andrew Howroyd, Nov 15 2017

From Roger Ford, Oct 17 2018: (Start)

a(n,k) = t(n+2,d) where t(n,d) is the number of different semi-meander arch depth listings with n top arches and with d the depth of the deepest embedded arch.

Examples:    /\         semi-meander with 5 top arches

            //\\   /\   2 arches are at depth=0 (no covering arches)

           ///\\\ //\\  2 arches are at depth=1 (1 covering arch)

      (0)(1)(2)         1 arch is at depth=2 (2 covering arches)

       2, 2, 1 is the listing for this t(5,2)

             /\      semi-meander with 5 top arches

            /  \     (0)(1)

     /\ /\ //\/\\     3, 2  is the listing for this t(5,1)

a(6,5) = t(8,5)= 3 {2,1,1,1,2,1; 2,1,2,1,1,1; 3,1,1,1,1,1} (End)

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.

Vladimir Shevelev, Several remarks on A088855, Seqfan thread, Nov 19 2017.

FORMULA

a(n, k) = binomial(floor(n'), floor(k'))*binomial(ceiling(n'), ceiling(k')), where n' = (n-1)/2, k' = (k-1)/2.

G.f.: 2u/(uv + sqrt(xyuv)) - 1, where x = 1+z+tz, y = 1+z-tz, u = 1-z+tz, v = 1-z-tz.

Triangle T(n,k), 0 <= k <= n, given by A101455 DELTA A056594 begins: 1; 0,1; 0,1,1; 0,1,1,1; 0,1,2,2,1; 0,1,2,4,2,1; 0,1,3,6,6,3,1; 0,1,3,9,9,9,3,1; ... - Philippe Deléham, Jan 03 2009

EXAMPLE

Triangle begins:

  1;

  1,1;

  1,1,1;

  1,2,2,1;

  1,2,4,2,1;

  1,3,6,6,3,1;

  1,3,9,9,9,3,1;

  1,4,12,18,18,12,4,1;

  1,4,16,24,36,24,16,4,1;

  1,5,20,40,60,60,40,20,5,1;

  1,5,25,50,100,100,100,50,25,5,1;

  1,6,30,75,150,200,200,150,75,30,6,1;

  1,6,36,90,225,300,400,300,225,90,36,6,1;

  1,7,42,126,315,525,700,700,525,315,126,42,7,1;

  1,7,49,147,441,735,1225,1225,1225,735,441,147,49,7,1;

  1,8,56,196,588,1176,1960,2450,2450,1960,1176,588,196,56,8,1;

  ...

a(6,2)=3 because we have UUUDDDUUUDDD, UUUUDDUUDDDD, UUUUUDUDDDDD, where

U=(1,1), D=(1,-1).

MATHEMATICA

T[n_, k_] := Binomial[Quotient[n-1, 2], Quotient[k-1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]];

Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 07 2018 *)

PROG

(PARI) T(n, k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ Andrew Howroyd, Nov 15 2017

CROSSREFS

Cf. A088459, A209612, A247644.

Column 2 is A008619, column 3 is A002620, column 4 is A028724, column 5 is A028723 and column 6 is A028725.

Sequence in context: A075402 A276696 A220777 * A034851 A172453 A172479

Adjacent sequences:  A088852 A088853 A088854 * A088856 A088857 A088858

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Nov 24 2003

EXTENSIONS

Keyword:tabl added Philippe Deléham, Jan 25 2010

STATUS

approved

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Last modified December 13 03:12 EST 2018. Contains 318081 sequences. (Running on oeis4.)