A191521 -id:A191521 - cp's OEIS frontend

A124428 Triangle, read by rows: T(n,k) = binomial(floor(n/2),k)*binomial(floor((n+1)/2),k).

1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 3, 1, 9, 9, 1, 1, 12, 18, 4, 1, 16, 36, 16, 1, 1, 20, 60, 40, 5, 1, 25, 100, 100, 25, 1, 1, 30, 150, 200, 75, 6, 1, 36, 225, 400, 225, 36, 1, 1, 42, 315, 700, 525, 126, 7, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 56, 588, 1960, 2450, 1176, 196, 8

Offset: 0

Paul D. Hanna, Oct 31 2006

Row sums form A001405, the central binomial coefficients: C(n,floor(n/2)). The eigenvector of this triangle is A124430.
T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having k peaks. Example: T(5,2)=3 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HUDUD, UDHUD, and UDUDH. - Emeric Deutsch, Jun 01 2011
From Emeric Deutsch, Jan 18 2013: (Start)
T(n,k) is the number of Dyck prefixes of length n having k peaks. Example: T(5,2)=3 because we have (UD)(UD)U, (UD)U(UD), and U(UD)(UD); the peaks are shown between parentheses.
T(n,k) is the number of Dyck prefixes of length n having k ascents and descents of length >= 2. Example: T(5,2)=3 because we have (UU)(DD)U, (UU)D(UU), and (UUU)(DD); the ascents and descents of length >= 2 are shown between parentheses. (End)
T(n,k) is the number of noncrossing partitions of [n] having n-k blocks, such that the nontrivial blocks are of type {a,b}, with a < = n/2 and b > n/2. Such partitions  have k nontrivial blocks, uniquely determined by the choice of k first elements among floor(n/2) elements, and the choice of k second elements among floor((n+1)/2) elements. Indeed, by planarity, any two blocs {a,b} and {c,d} satisfy a < c iff b > d. - Francesca Aicardi Nov 03 2022

			Triangle begins:
  1;
  1;
  1,   1;
  1,   2;
  1,   4,   1;
  1,   6,   3;
  1,   9,   9,   1;
  1,  12,  18,   4;
  1,  16,  36,  16,   1;
  1,  20,  60,  40,   5;
  1,  25, 100, 100,  25,   1;
  1,  30, 150, 200,  75,   6;
  1,  36, 225, 400, 225,  36,   1; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See paragraph 2.1.

Cf. A001405 (row sums), A056953, A026003, A124429 (antidiagonal sums), A124430 (eigenvector), A191521.

Columns = A002378, A006011, A006542, etc.

[[Binomial(Floor(n/2), k)*Binomial(Floor((n+1)/2),k): k in [0..Floor(n/2)]]: n in [0..15]]; // G. C. Greubel, Feb 24 2019

Table[Binomial[Floor[n/2], k]*Binomial[Floor[(n+1)/2], k], {n, 0, 15}, {k, 0, Floor[n/2]}]//Flatten (* G. C. Greubel, Feb 24 2019 *)

T(n,k)=binomial(n\2,k)*binomial((n+1)\2,k)

[[binomial(floor(n/2),k)*binomial(floor((n+1)/2),k) for k in (0..floor(n/2))] for n in (0..15)] # G. C. Greubel, Feb 24 2019

A056953(n) = Sum_{k=0..floor(n/2)} k!*T(n,k).

A026003(n) = Sum_{k=0..floor(n/2)} 2^k*T(n,k).

A191522 Number of valleys in all left factors of Dyck paths of length n. A valley is a (1,-1)-step followed by a (1,1)-step.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 20, 45, 105, 224, 504, 1050, 2310, 4752, 10296, 21021, 45045, 91520, 194480, 393822, 831402, 1679600, 3527160, 7113106, 14872858, 29953728, 62403600, 125550100, 260757900, 524190240, 1085822640, 2181340125, 4508102925, 9051563520, 18668849760

Offset: 0

Emeric Deutsch, Jun 05 2011

nonn

a(n+2) is also the sum of the maximum elements of each subset of [n]={1,...,n} with size floor((n+1)/2). For example for n=3 there are three subsets {1,2},{1,3},{2,3} and the sum of maximum values is 2+3+3=8. - Fabio Visonà, Aug 13 2023

			a(4)=3 because the total number of valleys in UDUD, UDUU, UUDD, UUDU, UUUD, and UUUU is 1+1+0+1+0+0=3; here U=(1,1), D=(1,-1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, Possible new formula for OEIS A191522

Cf. A191521.

q := sqrt(1-4*z^2): g := (2*((1-z-3*z^2+z^3)*q-1+z+5*z^2-3*z^3-4*z^4))/(z*q*(1-2*z-q)^2): gser := series(g, z = 0, 36): seq(coeff(gser, z, n), n = 0 .. 33);

CoefficientList[Series[(2*((1-x-3*x^2+x^3)*Sqrt[1-4*x^2]-1+x+5*x^2-3*x^3-4*x^4))/(x*Sqrt[1-4*x^2]*(1-2*x-Sqrt[1-4*x^2])^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

x='x+O('x^50); concat([0,0,0], Vec((2*((1-x-3*x^2+x^3)*sqrt(1-4*x^2)-1+x+5*x^2-3*x^3-4*x^4))/(x*sqrt(1-4*x^2)*(1-2*x-sqrt(1-4*x^2))^2))) \\ G. C. Greubel, Mar 26 2017

a(n) = Sum_{k>=0} k*A191521(n,k).
G.f.: 2*((1-z-3*z^2+z^3)*q-1+z+5*z^2-3*z^3-4*z^4)/(z*q*(1-2*z-q)^2), where q = sqrt(1-4*z^2).
a(n) ~ 2^(n-3/2)*sqrt(n)/sqrt(Pi). - Vaclav Kotesovec, Mar 21 2014
D-finite with recurrence -2*(n+1)*(n-3)*a(n) +(-5*n^2+29*n-6)*a(n-1) +2*(4*n+5)*(n-2)*a(n-2) +20*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
a(n) = floor((n-1)/2)*binomial(n-1,floor((n-1)/2)+1), n > 0. - Fabio Visonà, Aug 13 2023

cp's OEIS Frontend

A124428 Triangle, read by rows: T(n,k) = binomial(floor(n/2),k)*binomial(floor((n+1)/2),k).

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