A205810 -id:A205810 - cp's OEIS frontend

A051292 Whitney number of level n of the lattice of the ideals of the crown of size 2 n.

2, 1, 1, 4, 9, 21, 52, 127, 313, 778, 1941, 4863, 12228, 30837, 77967, 197574, 501657, 1275987, 3250618, 8292703, 21182509, 54169966, 138674031, 355343469, 911347684, 2339226871, 6008781637, 15445521202, 39728258103, 102248793573, 263306364822, 678411876729, 1748800672089

Offset: 0

Emanuele Munarini

A Chebyshev transform of the central binomial numbers A002426 under the mapping that takes g(x) to ((1-x^2)/(1+x^2))g(x/(1+x^2)). Starts 1,1,1,4,9,21,... - Paul Barry, Jan 31 2005

This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008

			a(3) = 4 because the ideals of size 3 of the crown C(3) = { x1 < x2 > x3 < x4 > x5 < x6 > x1 } are x1*x2*x3, x3*x4*x5, x1*x6*x5, x1*x3*x5.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alessandro Conflitti, On Whitney numbers of the Order Ideals of Generalized Fences and Crowns, arXiv:math/0505636 [math.CO], 2005.
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051291, A051286. Main diagonal of A205810.

f:= gfun:-rectoproc({n*(n-3)*a(n)-(2*n-1)*(n-3)*a(n-1)+(-n^2+4*n-5)*a(n-2)-(n-1)*(2*n-7)*a(n-3)+(n-1)*(n-4)*a(n-4) = 0, a(0) = 2, a(1) = 1, a(2) = 1, a(3) = 4},a(n),remember):
map(f, [$0..40]); # Robert Israel, Dec 06 2017
a := n -> `if`(n=0,2,2*add(((1+(-1)^(n-k)))*n*k*binomial((n+k)/2, k)^2*1/((n+k))^2, k=0..n)): seq(a(n), n=0..32); # Leonid Bedratyuk, Dec 07 2017

CoefficientList[Series[(1-x^2+Sqrt[1-2*x-x^2-2*x^3+x^4])/Sqrt[1-2*x-x^2-2*x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 05 2013 *)

x='x+O('x^66); Vec( (1-x^2+sqrt(1-2*x-x^2-2*x^3+x^4))/sqrt(1-2*x-x^2-2*x^3+x^4) ) \\ Joerg Arndt, May 04 2013

G.f.: (1 - t^2 + sqrt(1 - 2*t - t^2 - 2*t^3 + t^4))/sqrt(1 - 2*t - t^2 - 2*t^3 + t^4).
a(n) = sum{k=0..floor(n/2), (n/(n-k))C(n-k, k)*(-1)^k*sum{i=0..floor((n-2k)/2), C(n-2k, 2i)C(2i, i)}}; a(n)=sum{k=0..floor(n/2), (n/(n-k))C(n-k, k)*(-1)^k*A002426(n-2k)}. - Paul Barry, Jan 31 2005
Conjecture: n*(n-3)*a(n) - (2*n-1)*(n-3)*a(n-1) + (-n^2+4*n-5)*a(n-2) - (n-1)*(2*n-7)*a(n-3) + (n-1)*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 30 2012
Conjecture confirmed using the differential equation (2*x^2-x+2)*y(x) + (4*x^4-5*x^3-x^2+x-2)*y'(x) + (x^5-2*x^4-x^3-2*x^2+x)*y''(x) - 2*x^2 + x - 2 = 0 satisfied by the g.f. - Robert Israel, Dec 06 2017
a(n) ~ 5^(1/4)*((1+sqrt(5))/2)^(2*n)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 05 2013
a(n) = 2n Sum_{k=0..n}(1+(-1)^(n-k))*C((n+k)/2,k)^2*k/((n+k))^2 for n > 0. - Leonid Bedratyuk, Dec 06 2017

A097724 Triangle read by rows: T(n,k) is the number of left factors of Motzkin paths without peaks, having length n and endpoint height k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 4, 6, 6, 4, 1, 8, 13, 13, 10, 5, 1, 17, 28, 30, 24, 15, 6, 1, 37, 62, 69, 59, 40, 21, 7, 1, 82, 140, 160, 144, 105, 62, 28, 8, 1, 185, 320, 375, 350, 271, 174, 91, 36, 9, 1, 423, 740, 885, 852, 690, 474, 273, 128, 45, 10, 1, 978, 1728, 2102, 2077

Offset: 0

Emeric Deutsch, Sep 11 2004

Column 0 is A004148 (RNA secondary structure numbers).
This triangle appears identical to A191579 (apart from offsets). - Philippe Deléham, Jan 26 2014
Conjecture: the row reverse triangle is the triangle of connection constants for expressing the polynomial u(n,x+1) as a linear combination of the polynomials u(k,x), 0 <= k <= n, where u(n,x) = U(n,x/2) with U(n,x) the n-th Chebyshev polynomial of the second kind. An example is given below.  Cf. A205810. - Peter Bala, Jun 26 2025

			Triangle starts:
  1;
  1, 1;
  1, 2, 1;
  2, 3, 3, 1;
  4, 6, 6, 4, 1;
Row n has n+1 terms.
T(3,2)=3 because we have HUU, UHU and UUH, where U=(1,1) and H=(1,0).
Row 7: let u(n,x) = U(n,x/2). Then u(7,x+1) = u(7,x) + 7*u(6,x) + 21*u(5,x) + 40*u(4,x) + 59*u(3,x) + 69*u(2,x) + 62*u(1,x) + 37. - _Peter Bala_, Jun 26 2025

Alois P. Heinz, Rows n = 0..140, flattened
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
Naiomi Cameron and Everett Sullivan, Peakless Motzkin paths with marked level steps at fixed height, Discrete Mathematics 344.1 (2021): 112154.
Tian-Xiao He, A-sequences, Z-sequence, and B-sequences of Riordan matrices, Discrete Mathematics 343.3 (2020): 111718.
A. Nkwanta and A. Tefera, Curious Relations and Identities Involving the Catalan Generating Function and Numbers, Journal of Integer Sequences, 16 (2013), #13.9.5.
A. Panayotopoulos and P. Vlamos, Cutting Degree of Meanders, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From _N. J. A. Sloane_, Dec 29 2012

Cf. A004148, A191579, A091964 (row sums), A205810.

T:=proc(n,k) if k=n then 1 else (k+1)*sum(binomial(j,n-k-j)*binomial(j+k,n+1-j)/j,j=ceil((n-k+1)/2)..n-k) fi end: seq(seq(T(n,k),k=0..n),n=0..12); T:=proc(n,k) if k=n then 1 else (k+1)*sum(binomial(j,n-k-j)*binomial(j+k,n+1-j)/j,j=ceil((n-k+1)/2)..n-k) fi end: TT:=(n,k)->T(n-1,k-1): matrix(10,10,TT); # gives the sequence as a matrix

T[n_, k_] := T[n, k] = If[k==n, 1, (k+1)*Sum[Binomial[j, n-k-j]*Binomial[j +k, n+1-j]/j, {j, Ceiling[(n-k+1)/2], n-k}]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

T(n,k) = (k+1)*Sum_{j=ceiling((n-k+1)/2)..n-k} (C(j,n-k-j)*C(j+k,n+1-j)/j) for 0 <= k < n; T(n,n)=1.
G.f.: G/(1-tzG), where G = (1 - z + z^2 - sqrt(1 - 2z - z^2 - 2z^3 + z^4))/(2z^2) is the g.f. for the sequence A004148.
T(n,k) = T(n-1,k-1) + Sum_{j>=0} T(n-1-j,k+j), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 26 2014
Sum_{j=0..n-1} cos(2*Pi*k/3 + Pi/6)*T(n,k) = cos(Pi*n/2)*sqrt(3)/2 - cos(2*Pi*n/3 + Pi/6). - Leonid Bedratyuk, Dec 06 2017

cp's OEIS Frontend

A051292 Whitney number of level n of the lattice of the ideals of the crown of size 2 n.

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