A110122 -id:A110122 - cp's OEIS frontend

A110121 Triangle read by rows: T(n,k) (0 <= k <= floor(n/2)) is the number of Delannoy paths of length n, having k EE's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1).

1, 3, 12, 1, 53, 10, 247, 73, 1, 1192, 474, 17, 5897, 2908, 183, 1, 29723, 17290, 1602, 24, 152020, 100891, 12475, 342, 1, 786733, 581814, 90205, 3780, 31, 4111295, 3329507, 620243, 35857, 550, 1, 21661168, 18956564, 4114406, 307192, 7351, 38

Offset: 0

Emeric Deutsch, Jul 13 2005

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
Row n contains 1 + floor(n/2) terms.
Row sums are the central Delannoy numbers (A001850).

			T(2,0)=12 because, among the 13 (=A001850(2)) Delannoy paths of length 2, only NEEN has an EE crossing the line y=x.
Triangle begins:
    1;
    3;
   12,  1;
   53, 10;
  247, 73,  1;

Jinyuan Wang, Rows n = 0..50 of triangle, flattened
Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

Cf. A001850, A006318, A110122, A110123, A110127.

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/((1-z*R)^2-z-t*z^2*R^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 12 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form

nmax = 11; r := (1 - z - Sqrt[1 - 6*z + z^2])/2/z; g := 1/((1 - z*r)^2 - z - t*z^2*r^2); gser = Series[g, {z, 0, nmax}]; p[0] = 1; Do[ p[n] = Coefficient[ gser, z, n] , {n, 1, nmax}]; row[n_] := Table[ Coefficient[ t*p[n], t, k], {k, 1, 1 + Floor[n/2]}]; Flatten[ Table[ row[n], {n, 0, nmax}]] (* Jean-François Alcover, Dec 07 2011, after Maple *)

T(n,0) = A110122(n).
Sum_{k=0..floor(n/2)} k*T(n,k) = A110127(n).
G.f.: 1/((1 - zR)^2 - z - tz^2*R^2), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).

A132262 First term in a sum partition of the even-indexed Fibonacci numbers.

Original entry on oeis.org

1, 2, 7, 29, 130, 611, 2965, 14726, 74443, 381617, 1978582, 10355303, 54628201, 290148890, 1550177791, 8324934533, 44911554826, 243274479131, 1322555721037, 7213659006350, 39462884371891, 216470673634217, 1190382865461742, 6560913758341199

Offset: 0

Ph. Fahr and Claus Michael Ringel, Aug 19 2007

This is the number in the center of the 3-regular tree when the exceptional representations of the 3-Kronecker quiver, whose dimension vector is given by subsequent even-indexed Fibonacci numbers are drawn into the 3-regular tree (the universal cover of the quiver).

			a(3) = 29 because 377 = 29 + 6*18 + 24*6 + 96*1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Ph. Fahr and C. M. Ringel, A Partition Formula for Fibonacci Numbers, preprint, 2007.
Ph. Fahr and C. M. Ringel, A Partition Formula for Fibonacci Numbers, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.4
Philipp Fahr and Claus Michael Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers, Journal of Integer Sequences, Vol. 15 (2012), Article 12.2.1.
Pedro Fernando Fernández Espinosa and Agustín Moreno Cañadas, Homological Ideals as Integer Specializations of Some Brauer Configuration Algebras, arXiv:2104.00050 [math.RT], 2021.
Michael D. Hirschhorn, On Recurrences of Fahr and Ringel Arising in Graph Theory, Journal of Integer Sequences, Vol. 12 (2009), Article 09.6.8
Harris Kwong, On recurrences of Fahr and Ringel: an alternate approach, Fibonacci Quart. 48 (2010), no. 4, 363-365.
A. Moreno Canadas, P. F. Fernandez Espinoza, I. D. M. Gaviria, Categorification of some integer sequences via Kronecker Modules, JP J. Algebra, Number Theory and Applic. 38 (4) (2016) 339-347
H. Prodinger, Generating functions related to partition formulas for Fibonacci Numbers, JIS 11 (2008) 08.1.8.

Cf. A110122.

a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 29][n+1],
       ((13*n-9)*a(n-1) -(44*n-66)*a(n-2)
       +(13*n-30)*a(n-3) -(n-3)*a(n-4))/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 19 2013

a[n_] := a[n] = If[n<4, {1, 2, 7, 29}[[n+1]], ((13*n-9)*a[n-1] - (44*n-66)*a[n-2] + (13*n-30)*a[n-3] - (n-3)*a[n-4])/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 07 2016, after Alois P. Heinz *)

lista(nn) = my(q = qq + O(qq^nn)); gf = (3*sqrt(1-6*q+q^2) -(1+q))/(2*(1-7*q+q^2)); Vec(gf) \\ Michel Marcus, Jun 19 2013

G.f.: (3*sqrt(1-6*q+q^2)-(1+q))/(2*(1-7*q+q^2)) = 1 +2q +7q^2 +29q^3 +130q^4 + ... . Michael David Hirschhorn, Sep 03 2009
a(n) ~ 3*sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^(n+1) / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 08 2014
D-finite with recurrence n*a(n) + (-13*n+9)*a(n-1) + 22*(2*n-3)*a(n-2) + (-13*n+30)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Aug 28 2015

A197956 Even-index Fibonacci partition triangle read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 7, 1, 5, 12, 1, 6, 18, 29, 1, 7, 25, 53, 1, 8, 33, 85, 130, 1, 9, 42, 126, 247, 1, 10, 52, 177, 414, 611, 1, 11, 63, 239, 642, 1192, 1, 12, 75, 313, 943, 2062, 2965, 1, 13, 88, 400, 1330, 3306, 5897, 1, 14, 102, 501, 1817, 5023, 10447, 14726

Offset: 1

Claus Michael Ringel, Oct 24 2011

These are the entries of a triangle which starts
           1,
           1,
       1,  2,
       1,  3,
   1,  4,  7,
   1,  5, 12,
1, 6, 18, 29,...
The numbers d(i,n) in the row with index n are recursively defined for 0 <= n and 0 <= i <= n/2, by d(0,n) = 1 for all n, and d(i,n) = 2d(i-1,n-1) + d(i,n-1) - d(i-1,n-2) for 0 < i < n/2, and d(i,2i) = 3d(i-1,n-1) - d(i-1,n-2).
The numbers d(i,n-1) and d(i,n) form the dimension vector of the Fibonacci modules P(n), these are indecomposable quiver representations of the 3-regular tree with bipartite orientation.
A linear combination of the row n (with coefficients either 1 or of the form 3*2^t) gives a partition of the Fibonacci number f_{2n+2}, see A000045 and A001906.
The sequence A177011 is obtained by reading the rows with even index from the right.
The sequence A177020 is obtained by reading the rows with odd index from the right.
The sequence of the entries in the last column of the triangle, with even row index is recorded as A132262, the sequence of the entries in the last column of the triangle, with odd row index is A110122 (Number of Delannoy paths of length n with no EE's crossing the line y=x)
The sequence A197957 is obtained by taking differences of pairs of numbers in neighboring rows of the triangle.

Philipp Fahr and Claus Michael Ringel, The Fibonacci partition triangles, http://arxiv.org/abs/1109.2849
P. Fahr, C. M. Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers, J. Int. Seq. 15 (2012) # 12.2.1

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A110121 Triangle read by rows: T(n,k) (0 <= k <= floor(n/2)) is the number of Delannoy paths of length n, having k EE's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1).

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A132262 First term in a sum partition of the even-indexed Fibonacci numbers.

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A197956 Even-index Fibonacci partition triangle read by rows.

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