OFFSET
1,4
COMMENTS
From Gary W. Adamson, Apr 24 2005: (Start)
Let A be the array
1, 0, 0, 0, 0, 0, ...
0, 1, 0, 0, 0, 0, ...
1, 0, 1, 0, 0, 0, ...
0, 2, 0, 1, 0, 0, ...
1, 0, 3, 0, 1, 0, ...
0, 3, 0, 4, 0, 1, ...
...
where columns are bin(n,k) with alternating zeros. (Row sums = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...(Fibonacci numbers).) Let P = infinite lower triangular Pascal triangle matrix (A007318). Form P * A: this gives the rows of the present sequence. [Comment corrected by Philippe Deléham, Dec 09 2008] (End)
T(n,k) is the number of nondecreasing Dyck paths of semilength n, having height of rightmost peak equal to k. Example: T(4,1)=4 because we have UDUDUDUD, UDUUDDUD, UUDDUDUD and UUUDDDUD, where U=(1,1) and D=(1,-1). Sum of row n = Fibonacci(2n-1) (A001519). Basically the same as A062110.
T(n,k) is the number of permutations of [n] with length n-k that avoid the patterns 321 and 3412. - Bridget Tenner, Sep 28 2005
T(2*n-1,n)/n = A001003(n-1) (little Schroeder numbers). Proof with Lagrange inversion of inverse of g.f. of A001003.
Row sums = odd-indexed Fibonacci numbers.
Diagonal sums: A077998. - Philippe Deléham, Nov 16 2008
Riordan matrix ((1-x)/(1-2x),(x-x^2)/(1-2x)). - Emanuele Munarini, Mar 22 2011
T(n,k) is the number of ideals in the fence Z(2n-1) with n-k elements of rank 0. - Emanuele Munarini, Mar 22 2011
Triangle T(n,k), 1 <= k <= n, read by rows, given by (0,1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 30 2011
T(n,k) is the number of permutations of [n] for which k is equal to both the length and reflection length. - Bridget Tenner, Feb 22 2012
REFERENCES
V. E. Hoggatt, Jr. and Marjorie Bicknell, editors: "A Primer for the Fibonacci Numbers", 1970, p. 87.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022).
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, Juana Sendra, and Steven E. Thornton, Bohemian Upper Hessenberg Toeplitz Matrices, arXiv:1809.10664 [cs.SC], 2018.
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, and Juana Sendra, Upper Hessenberg and Toeplitz Bohemians, arXiv:1907.10677 [cs.SC], 2019.
Emeric Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.9.3.
E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
T. K. Petersen and B. E. Tenner, The depth of a permutation, arXiv:1202.4765 [math.CO], 2012-2014..
M. Pétréolle, Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups, arXiv preprint arXiv:1403.1130 [math.GR], 2014.
Bridget Eileen Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.
FORMULA
T(n,k) = sum(binomial(k+j, k-1)*binomial(n-k-1, j), j=0..n-k-1) (0<=k<=n). (This appears to be incorrect. - Emanuele Munarini, Mar 22 2011)
G.f.: t*z*(1-z)/(1 - 2*z - t*z*(1-z)).
From Emanuele Munarini, Mar 22 2011: (Start)
T(n,k) = Sum_{i=0..n-k} binomial(k+1,i)*binomial(n-i,k)*(-1)^i*2^(n-k-i).
T(n,k) = Sum_{i=0..n-k} binomial(k+1,i)*M(i,n-k-i)*2^(n-k-i), where M(n,k) = n*(n+1)*(n+2)*...*(n+k-1)/k!. (End)
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 30 2013
EXAMPLE
Triangle begins:
1;
1, 1;
2, 2, 1;
4, 5, 3, 1;
8, 12, 9, 4, 1;
16, 28 25, 14, 5, 1;
32, 64, 66, 44, 20, 6, 1;
64, 144, 168, 129, 70, 27, 7, 1;
...
From Paul Barry, Apr 18 2010: (Start)
Production matrix is
1, 1;
1, 1, 1;
0, 1, 1, 1;
-1, 0, 1, 1, 1;
0, -1, 0, 1, 1, 1;
2, 0, -1, 0, 1, 1, 1;
0, 2, 0, -1, 0, 1, 1, 1;
-5, 0, 2, 0, -1, 0, 1, 1, 1;
0, -5, 0, 2, 0, -1, 0, 1, 1, 1;
14, 0, -5, 0, 2, 0, -1, 0, 1, 1, 1; (End)
MAPLE
T:=proc(n, k) if k<n-1 then sum(binomial(k+j, k-1)*binomial(n-k-1, j), j=0..n-k-1) elif k=n-1 then n-1 elif k=n then 1 else 0 fi end: for n from 1 to 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
MATHEMATICA
t[n_, k_] := 2^(n-2*k-1)*Binomial[n, k]*Hypergeometric2F1[-k-1, -k, -n, -1]; t[n_, n_] = 1; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)
PROG
(Maxima) create_list(sum(binomial(k+1, i)*binomial(n-i, k)*(-1)^i*2^(n-k-i), i, 0, n-k), n, 0, 8, k, 0, n); /* Emanuele Munarini, Mar 22 2011 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Apr 25 2005
EXTENSIONS
Entry revised by N. J. A. Sloane, Apr 27 2007
STATUS
approved