cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Stefano Spezia, Apr 14 2024

Keywords

Examples

			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.).

Links

Crossrefs

Cf. A371963, A371964.
Cf. A002054, A275289.

Programs

  • Maple
    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
  • Mathematica
    CoefficientList[Series[(1-3x-(1-x)Sqrt[1-4x])/(2(1-x) Sqrt[1-4x]),{x,0,28}],x]
  • Python
    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

Formula

G.f.: (1 - 3*x - (1 - x)*sqrt(1 - 4*x))/(2*(1 - x)*sqrt(1 - 4*x)).
a(n) = Sum_{i=1..n-1} binomial(2*(n-i)-1,n-i-2).
a(n) ~ 2^(2*n)/(6*sqrt(Pi*n)).
a(n)/A371963(n) ~ 1.
a(n) - a(n-1) = A002054(n-2).
From Mélika Tebni, Jun 15 2024: (Start)
E.g.f.: (exp(2*x)*BesselI(0,2*x)-1)/2 - exp(x)*Integral_{x=-oo..oo} BesselI(0,2*x)*exp(x) dx.
a(n) = binomial(2*n,n)*(1/2 + hypergeom([1,n+1/2],[n+1],4)) + i/sqrt(3) - 0^n/2.
a(n) = (3*A106191(n) + A006134(n) + 4*0^n) / 8.
a(n) = A281593(n) - (A000984(n) + 0^n) / 2. (End)
Binomial transform of A275289. - Alois P. Heinz, Jun 20 2025

A371963 a(n) is the sum of all valleys in the set of Catalan words of length n.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 44, 209, 924, 3927, 16303, 66691, 270181, 1087371, 4356131, 17394026, 69289961, 275543036, 1094352236, 4342295396, 17218070066, 68239187876, 270351828476, 1070824260326, 4240695090452, 16792454677874, 66492351226050, 263285419856250, 1042540731845950
Offset: 0

Views

Author

Stefano Spezia, Apr 14 2024

Keywords

Examples

			a(4) = 1 because there is 1 Catalan word of length 4 with one valley: 0101.
a(5) = 8 because there are 8 Catalan words of length 5 with one valley: 00101, 01010, 01011, 01012, 01101, 01201, and 01212 (see Figure 9 at p. 14 in Baril et al.).

Links

Crossrefs

Cf. A371964, A371965.
Cf. A003516.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, 0,
      a(n-1)+binomial(2*n-3, n-4))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Apr 15 2024
  • Mathematica
    CoefficientList[Series[(1 - 5x+5x^2-(1-3x+x^2)Sqrt[1-4x])/(2(1-x)x Sqrt[1-4x]),{x,0,28}],x]
  • Python
    from math import comb
def A371963(n): return sum(comb((n-i<<1)-3,n-i-4) for i in range(n-3)) # Chai Wah Wu, Apr 15 2024

Formula

G.f.: (1-5*x+5*x^2-(1-3*x+x^2)*sqrt(1-4*x))/(2*(1-x)*x*sqrt(1-4*x)).
a(n) = Sum_{i=1..n-1} binomial(2*(n-i)-1,n-i-3).
a(n) ~ 2^(2*n)/(6*sqrt(Pi*n)).
a(n) - a(n-1) = A003516(n-2).

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

Views

Author

Stefano Spezia, May 15 2024

Keywords

Examples

			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.

Links

  • Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 16-17.

Crossrefs

Cf. A007051 (row sums), A371963, A371964, A372878.

Programs

  • Mathematica
    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]}]]]

Formula

G.f.: x*(1 - 2*x)*(1 - 2*x + 2*x^2 - x^2*y)/((1 - x)*(1 - 5*x + 8*x^2 - 5*x^3 - x^2*y + 2x^3*y)).
Sum_{k>=0} T(n,k) = A007051(n-1).

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

Views

Author

Stefano Spezia, May 15 2024

Keywords

Comments

The g.f. listed in Baril et al. has a mistake in the numerator: the factor (1 + 2*x) should be (1 - 2*x).

Links

Crossrefs

Cf. A261064, A371963, A371964, A372875.

Programs

  • Mathematica
    LinearRecurrence[{9,-30,46,-33,9},{1,7,33,133,496},28]

Formula

From Baril et al.: (Start)
G.f.: x^4*(1 - 2*x)/((1 - 3*x)^2*(1 - x)^3).
a(n) = (3^n*(2*n - 5) - 18*n^2 + 54*n - 27)/144. (End)
E.g.f.: (32 + exp(3*x)*(6*x - 5) - 9*exp(x)*(2*x^2 - 4*x + 3))/144.
a(n) - a(n-1) = A261064(n-3).
Showing 1-4 of 4 results.

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