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%I #64 Apr 06 2023 02:35:36
%S 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,
%T 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,
%U 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
%N Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.
%C Rows 2, 4, 6, ... give A088459.
%C Diagonal sums are in A088518(n-1). - _Philippe Deléham_, Jan 04 2009
%C Row sums are in A001405(n). - _Philippe Deléham_, Jan 04 2009
%C 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
%C Also, number of symmetric noncrossing partitions of an n-set with k blocks. - _Andrew Howroyd_, Nov 15 2017
%C From _Roger Ford_, Oct 17 2018: (Start)
%C T(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.
%C Examples: /\ semi-meander with 5 top arches
%C //\\ /\ 2 arches are at depth=0 (no covering arches)
%C ///\\\ //\\ 2 arches are at depth=1 (1 covering arch)
%C (0)(1)(2) 1 arch is at depth=2 (2 covering arches)
%C 2, 2, 1 is the listing for this t(5,2)
%C /\ semi-meander with 5 top arches
%C / \ (0)(1)
%C /\ /\ //\/\\ 3, 2 is the listing for this t(5,1)
%C 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)
%H Andrew Howroyd, <a href="/A088855/b088855.txt">Table of n, a(n) for n = 1..1275</a>
%H Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, <a href="https://arxiv.org/abs/2010.11157">Refined Catalan and Narayana cyclic sieving</a>, arXiv:2010.11157 [math.CO], 2020.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H Hyunsoo Cho, JiSun Huh, and Jaebum Sohn, <a href="https://arxiv.org/abs/2001.06651">The (s, s + d, ..., s + pd)-core partitions and the rational Motzkin paths</a>, arXiv:2001.06651 [math.CO], 2020.
%H Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015-2016.
%H Johann Cigler, <a href="https://arxiv.org/abs/2103.01652">Pascal triangle, Hoggatt matrices, and analogous constructions</a>, arXiv:2103.01652 [math.CO], 2021.
%H Nicolas Crampe, Julien Gaboriaud, and Luc Vinet, <a href="https://arxiv.org/abs/2105.01086">Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series</a>, arXiv:2105.01086 [math.RT], 2021.
%H L. Poulain d'Andecy, <a href="https://arxiv.org/abs/2304.00850">Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics</a>, arXiv:2304.00850 [math.RT], 2023. See p. 64.
%H Vladimir Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-November/018114.html">Several remarks on A088855</a>, Seqfan thread, Nov 19 2017.
%F T(n, k) = binomial(floor(n'), floor(k'))*binomial(ceiling(n'), ceiling(k')), where n' = (n-1)/2, k' = (k-1)/2.
%F G.f.: 2*u/(u*v + sqrt(x*y*u*v)) - 1, where x = 1+z+t*z, y = 1+z-t*z, u = 1-z+t*z, v = 1-z-t*z.
%F 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
%F From _G. C. Greubel_, Apr 08 2022: (Start)
%F T(n, n-k+1) = T(n, k).
%F T(2*n-1, n) = A018224(n-1), n >= 1.
%F T(2*n, n) = A005566(n-1), n >= 1. (End)
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 2, 4, 2, 1;
%e 1, 3, 6, 6, 3, 1;
%e 1, 3, 9, 9, 9, 3, 1;
%e 1, 4, 12, 18, 18, 12, 4, 1;
%e 1, 4, 16, 24, 36, 24, 16, 4, 1;
%e 1, 5, 20, 40, 60, 60, 40, 20, 5, 1;
%e 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1;
%e 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1;
%e 1, 6, 36, 90, 225, 300, 400, 300, 225, 90, 36, 6, 1;
%e 1, 7, 42, 126, 315, 525, 700, 700, 525, 315, 126, 42, 7, 1;
%e 1, 7, 49, 147, 441, 735, 1225, 1225, 1225, 735, 441, 147, 49, 7, 1;
%e 1, 8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1;
%e ...
%e a(6,2)=3 because we have UUUDDDUUUDDD, UUUUDDUUDDDD, UUUUUDUDDDDD, where
%e U=(1,1), D=(1,-1).
%t T[n_, k_] := Binomial[Quotient[n-1, 2], Quotient[k-1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]];
%t Table[T[n, k], {n,13}, {k,n}]//Flatten (* _Jean-François Alcover_, Jun 07 2018 *)
%o (PARI) T(n,k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ _Andrew Howroyd_, Nov 15 2017
%o (Magma) [(&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Apr 08 2022
%o (Sage)
%o def A088855(n,k): return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
%o flatten([[A088855(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Apr 08 2022
%Y Cf. A005566, A018224, A056594, A084938, A088459, A101455, A209612, A247644.
%Y Cf. A001405 (row sums), A088459, A088518 (diagonal sums).
%Y Column 2 is A008619, column 3 is A002620, column 4 is A028724, column 5 is A028723, column 6 is A028725, column 7 is A331574.
%K nonn,tabl
%O 1,8
%A _Emeric Deutsch_, Nov 24 2003
%E Keyword:tabl added _Philippe Deléham_, Jan 25 2010
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