A275289 -id:A275289 - cp's OEIS frontend

A275281 Number T(n,k) of set partitions of [n] with symmetric block size list of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 7, 0, 1, 0, 1, 10, 19, 13, 3, 1, 0, 1, 0, 56, 0, 22, 0, 1, 0, 1, 35, 160, 171, 86, 34, 4, 1, 0, 1, 0, 463, 0, 470, 0, 50, 0, 1, 0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1, 0, 1, 0, 3874, 0, 10299, 0, 2160, 0, 95, 0, 1

Offset: 0

Alois P. Heinz, Jul 21 2016

			T(4,2) = 3: 12|34, 13|24, 14|23.
T(5,3) = 7: 12|3|45, 13|2|45, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34.
T(6,4) = 13: 12|3|4|56, 13|2|4|56, 1|23|45|6, 1|23|46|5, 14|2|3|56, 1|24|35|6, 1|24|36|5, 1|25|34|6, 1|26|34|5, 15|2|3|46, 1|25|36|4, 1|26|35|4, 16|2|3|45.
T(7,5) = 22: 12|3|4|5|67, 13|2|4|5|67, 1|23|4|56|7, 1|23|4|57|6, 14|2|3|5|67, 1|24|3|56|7, 1|24|3|57|6, 1|2|345|6|7, 1|2|346|5|7, 1|2|347|5|6, 15|2|3|4|67, 1|25|3|46|7, 1|25|3|47|6, 1|2|356|4|7, 1|2|357|4|6, 1|26|3|45|7, 1|27|3|45|6, 16|2|3|4|57, 1|26|3|47|5, 1|2|367|4|5, 1|27|3|46|5, 17|2|3|4|56.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   0,    1;
  0, 1,   3,    2,    1;
  0, 1,   0,    7,    0,    1;
  0, 1,  10,   19,   13,    3,    1;
  0, 1,   0,   56,    0,   22,    0,   1;
  0, 1,  35,  160,  171,   86,   34,   4,  1;
  0, 1,   0,  463,    0,  470,    0,  50,  0, 1;
  0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition of a set

Columns k=0-1,3,5,7,9 give: A000007, A000012 for n>0, A275289, A275290, A275291, A275292.
Bisections of columns k=2,4,6,8,10 give: A001700(n-1) for n>0, A275293, A275294, A275295, A275296.
Row sums give A275282.
T(n,A004525(n)) gives A305197.
T(2n,n) gives A275283.
T(2n+1,A109613(n)) gives A305198.
T(n,n) gives A000012.
T(n+3,n+1) gives A002623.

b:= proc(n, s) option remember; expand(`if`(n>s,
      binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
      b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..12);

b[n_, s_] := b[n, s] = Expand[If[n>s, Binomial[n-1, n-s-1]*x, 1] + Sum[ Binomial[n-1, j-1]*b[n-j, s+j]*Binomial[s+j-1, j-1], {j, 1, (n-s)/2} ]*x^2]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

T(n,k) = 0 if n is odd and k is even.

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

Stefano Spezia, Apr 14 2024

nonn

			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.

Cf. A371963, A371964.

Cf. A002054, A275289.

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

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

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

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

cp's OEIS Frontend

A275281 Number T(n,k) of set partitions of [n] with symmetric block size list of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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