A254632
Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.
Original entry on oeis.org
1, 4, 2, 16, 16, 5, 64, 96, 60, 14, 256, 512, 480, 224, 42, 1024, 2560, 3200, 2240, 840, 132, 4096, 12288, 19200, 17920, 10080, 3168, 429, 16384, 57344, 107520, 125440, 94080, 44352, 12012, 1430, 65536, 262144, 573440, 802816, 752640, 473088, 192192, 45760, 4862
Offset: 0
[ 1]
[ 4, 2]
[ 16, 16, 5]
[ 64, 96, 60, 14]
[ 256, 512, 480, 224, 42]
[1024, 2560, 3200, 2240, 840, 132]
[4096, 12288, 19200, 17920, 10080, 3168, 429]
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h := n -> simplify(hypergeom([3/2, -n], [3], -x)):
seq(print(seq(4^n*coeff(h(n), x, k), k=0..n)), n=0..9);
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T[n_, k_] := 4^(n-k) Binomial[n, k] CatalanNumber[k+1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-François Alcover, Jun 28 2019 *)
-
A254632 = lambda n,k: (4)^(n-k)*binomial(n,k)*catalan_number(k+1)
for n in range(7): [A254632(n,k) for k in (0..n)]
A164991
Number of triangular involutions of n. A triangular involution is a square involution with at most three faces.
Original entry on oeis.org
1, 1, 3, 6, 13, 26, 54, 108, 221, 442, 898, 1796, 3634, 7268, 14668, 29336, 59101, 118202, 237834, 475668, 956198, 1912396, 3841588, 7683176, 15425138, 30850276, 61908564, 123817128, 248377156, 496754312
Offset: 1
Simone Rinaldi (rinaldi(AT)unisi.it), Sep 04 2009
- F. Disanto, A. Frosini, S. Rinaldi, Square Involutions, Proceedings of Permutation Patterns, July, 13-17 2009, Florence.
- T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, 19th Conference on Formal Power Series and Algebraic Combinatorics, Tianjin, China, July 2-6, 2007.
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Join[{1},Table[2^(n-1)-Binomial[n-2,Floor[(n-2)/2]],{n,2,30}]] (* Harvey P. Dale, Dec 26 2015 *)
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a(n) = 2^(n-1) - binomial(n-2, (n-2)\2) \\ Michel Marcus, May 27 2013
A276418
Starting a random walk on Z at 0 triangle T(j,k) gives the number of paths of length 2*j returning to 0 exactly k times.
Original entry on oeis.org
1, 2, 2, 6, 6, 4, 20, 20, 16, 8, 70, 70, 60, 40, 16, 252, 252, 224, 168, 96, 32, 924, 924, 840, 672, 448, 224, 64, 3432, 3432, 3168, 2640, 1920, 1152, 512, 128, 12870, 12870, 12012, 10296, 7920, 5280, 2880, 1152, 256, 48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512, 184756, 184756, 175032, 155584, 128128, 96096, 64064, 36608, 16896, 5632, 1024
Offset: 0
Triangle T(j,k) begins:
1
2, 2
6, 6, 4
20, 20, 16, 8
70, 70, 60, 40, 16
252, 252, 224, 168, 96, 32
924, 924, 840, 672, 448, 224, 64
3432, 3432, 3168, 2640, 1920, 1152, 512, 128
12870, 12870, 12012, 10296, 7920, 5280, 2880, 1152, 256
48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512
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Flat(List([0..10],j->List([0..j],k->2^k*Binomial(2*j-k,j-k)))); # Muniru A Asiru, May 18 2018
Showing 1-3 of 3 results.
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