OFFSET
1,8
COMMENTS
Rows 2, 4, 6, ... give A088459.
Diagonal sums are in A088518(n-1). - Philippe Deléham, Jan 04 2009
Row sums are in A001405(n). - 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)
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.
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
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Hyunsoo Cho, JiSun Huh, and Jaebum Sohn, The (s, s + d, ..., s + pd)-core partitions and the rational Motzkin paths, arXiv:2001.06651 [math.CO], 2020.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Nicolas Crampe, Julien Gaboriaud, and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.
Vladimir Shevelev, Several remarks on A088855, Seqfan thread, Nov 19 2017.
FORMULA
T(n, k) = binomial(floor(n'), floor(k'))*binomial(ceiling(n'), ceiling(k')), where n' = (n-1)/2, k' = (k-1)/2.
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.
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
From G. C. Greubel, Apr 08 2022: (Start)
T(n, n-k+1) = T(n, k).
T(2*n-1, n) = A018224(n-1), n >= 1.
T(2*n, n) = A005566(n-1), n >= 1. (End)
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, 13}, {k, 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
(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
(Sage)
def A088855(n, k): return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A088855(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 08 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Nov 24 2003
EXTENSIONS
Keyword:tabl added Philippe Deléham, Jan 25 2010
STATUS
approved