A371965
a(n) is the sum of all peaks in the set of Catalan words of length n.
Original entry on oeis.org
0, 0, 0, 1, 6, 27, 111, 441, 1728, 6733, 26181, 101763, 395693, 1539759, 5997159, 23381019, 91244934, 356427459, 1393585779, 5453514729, 21358883439, 83718027429, 328380697629, 1288947615849, 5062603365999, 19896501060225, 78239857877649, 307831771279549, 1211767933187601
Offset: 0
a(3) = 1 because there is 1 Catalan word of length 3 with one peak: 010.
a(4) = 6 because there are 6 Catalan words of length 4 with one peak: 0010, 0100, 0101, 0110, 0120, and 0121 (see Figure 10 at p. 19 in Baril et al.).
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See Corollary 4.7, p. 19.
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a:= proc(n) option remember; `if`(n<3, 0,
a(n-1)+binomial(2*n-3, n-3))
end:
seq(a(n), n=0..28); # Alois P. Heinz, Apr 15 2024
# Second Maple program:
A371965 := series((exp(2*x)*BesselI(0,2*x)-1)/2-exp(x)*(int(BesselI(0,2*x)*exp(x), x)), x = 0, 29):
seq(n!*coeff(A371965, x, n), n = 0 .. 28); # Mélika Tebni, Jun 15 2024
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CoefficientList[Series[(1-3x-(1-x)Sqrt[1-4x])/(2(1-x) Sqrt[1-4x]),{x,0,28}],x]
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from math import comb
def A371965(n): return sum(comb((n-i<<1)-3,n-i-3) for i in range(n-2)) # Chai Wah Wu, Apr 15 2024
A371964
a(n) is the sum of all symmetric valleys in the set of Catalan words of length n.
Original entry on oeis.org
0, 0, 0, 0, 1, 7, 35, 155, 650, 2652, 10660, 42484, 168454, 665874, 2627130, 10353290, 40775045, 160534895, 631970495, 2487938015, 9795810125, 38576953505, 151957215305, 598732526105, 2359771876175, 9303298456451, 36688955738099, 144732209103699, 571117191135799
Offset: 0
a(4) = 1 because there is 1 Catalan word of length 4 with one symmetric valley: 0101.
a(5) = 7 because there are 7 Catalan words of length 5 with one symmetric valley: 00101, 01001, 01010, 01011, 01012, 01101, and 01212 (see example at p. 16 in Baril et al.).
- Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See Corollary 4.7, pp. 16-17.
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a:= proc(n) option remember; `if`(n<4, 0,
a(n-1)+binomial(2*n-4, n-4))
end:
seq(a(n), n=0..28); # Alois P. Heinz, Apr 15 2024
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CoefficientList[Series[(1-4x+2x^2-(1-2x)Sqrt[1-4x])/(2(1-x) Sqrt[1-4x]),{x,0,29}],x]
-
from math import comb
def A371964(n): return sum(comb((n-i<<1)-4,n-i-4) for i in range(n-3)) # Chai Wah Wu, Apr 15 2024
A372875
Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k symmetric valleys, with k >= 0.
Original entry on oeis.org
1, 2, 5, 13, 1, 34, 7, 90, 31, 1, 242, 113, 10, 659, 375, 59, 1, 1808, 1189, 271, 13, 4977, 3686, 1082, 96, 1, 13715, 11284, 3976, 534, 16, 37798, 34239, 13887, 2507, 142, 1, 104154, 103115, 46949, 10555, 929, 19, 286960, 308452, 155200, 41324, 5028, 197, 1
Offset: 1
The irregular triangle begins:
1;
2;
5;
13, 1;
34, 7;
90, 31, 1;
242, 113, 10;
659, 375, 59, 1;
1808, 1189, 271, 13;
4977, 3686, 1082, 96, 1;
...
T(5,1) = 7 since there are 7 flattened Catalan words of length 5 with one symmetric valley: 00101, 01001, 01010, 01011, 01012, 01101, and 01212.
- Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 16-17.
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T[n_,k_]:=SeriesCoefficient[x(1-2x)(1-2x+2x^2-x^2y)/((1-x)(1-5x+8x^2-5x^3-x^2y+2x^3y)),{x,0,n},{y,0,k}];Join[{1},Flatten[ Table[T[n,k],{n,14},{k,0,Floor[(n-2)/2]}]]]
A372878
a(n) is the sum of all symmetric valleys in the set of flattened Catalan words of length n.
Original entry on oeis.org
1, 7, 33, 133, 496, 1770, 6142, 20902, 70107, 232489, 763927, 2491107, 8071234, 26007364, 83402988, 266351548, 847482277, 2687729595, 8499036925, 26804655025, 84336597636, 264777690382, 829636763338, 2594821366338, 8102197327711, 25259791668925, 78638974063827
Offset: 4
- Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 18.
- Index entries for linear recurrences with constant coefficients, signature (9,-30,46,-33,9).
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LinearRecurrence[{9,-30,46,-33,9},{1,7,33,133,496},28]
Showing 1-4 of 4 results.
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