A049600 Array T read by diagonals; T(i,j) is the number of paths from (0,0) to (i,j) consisting of nonvertical segments (x(k),y(k))-to-(x(k+1),y(k+1)) such that 0 = x(1) < x(2) < ... < x(n-1) < x(n)=i, 0 = y(1) <= y(2) <= ... <= y(n-1) <= y(n)=j, for i >= 0, j >= 0.

0, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 4, 8, 8, 0, 1, 5, 13, 20, 16, 0, 1, 6, 19, 38, 48, 32, 0, 1, 7, 26, 63, 104, 112, 64, 0, 1, 8, 34, 96, 192, 272, 256, 128, 0, 1, 9, 43, 138, 321, 552, 688, 576, 256, 0, 1, 10, 53, 190, 501, 1002, 1520, 1696, 1280, 512

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• easy

Comments

Essentially array A059576 divided by sequence A011782.

[Hetyei] calls a variant of this array (omitting the initial row of zeros) the asymmetric Delannoy numbers and shows how they arise in certain lattice path enumeration problems and a face enumeration problem associated to Jacobi polynomials. - Peter Bala, Oct 29 2008

Essentially triangle in A208341. - Philippe Deléham, Mar 23 2012

T(n+k,n) is the dot product of a vector from the n-th row of Pascal's triangle A007318 with a vector created by the first n+1 values evaluated from the cycle index of symmetry group S(k). Example: T(4+3,4) = T(7,4) = {1,4,6,4,1}.{1,4,10,20,35} = 192. - Richard Turk, Sep 21 2017

The formula T(n,k) = Sum_{r=0..n-1} C(k+r,r)*C(n-1,r) (Paul D. Hanna, Oct 06 2006) counts the paths of the title by number, r, of interior vertices in the path. - David Callan, Nov 25 2021

Examples

			Diagonals (each starting on row 1): {0}; {0,1}; {0,1,2}; ...
Array begins:
    0     0     0     0     0     0     0     0     0     0     0 ...
    1     1     1     1     1     1     1     1     1     1     1 ...
    2     3     4     5     6     7     8     9    10    11    12 ...
    4     8    13    19    26    34    43    53    64    76    89 ...
    8    20    38    63    96   138   190   253   328   416   518 ...
   16    48   104   192   321   501   743  1059  1462  1966  2586 ...
   32   112   272   552  1002  1683  2668  4043  5908  8378 11584 ...
   64   256   688  1520  2972  5336  8989 14407 22180 33028 47818 ...
Triangle begins:
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3,  4;
  0, 1, 4,  8,  8;
  0, 1, 5, 13, 20,  16;
  0, 1, 6, 19, 38,  48,  32;
  0, 1, 7, 26, 63, 104, 112, 64;
  ...
(1, 0, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) where DELTA is the operator defined in A084938 begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  4,  0;
  1, 4,  8,  8,   0;
  1, 5, 13, 20,  16,   0;
  1, 6, 19, 38,  48,  32,  0;
  1, 7, 26, 63, 104, 112, 64, 0;
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of table, flattened
• David Callan, Some bijections for lattice paths, arXiv:2112.05241 [math.CO], 2021.
• David Callan, A bijection for Delannoy paths, arXiv:2202.04649 [math.CO], 2022.
• R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
• R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
• Robert Cori and Gabor Hetyei, Genus one partitions, in 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AT, pp. 333-344, .
• Sergio Falcon, On the complex k-Fibonacci numbers, Cogent Mathematics, (2016), 3: 1201944. See Table 1.
• G. Hetyei, Central Delannoy numbers, Legendre polynomials and a balanced join operation preserving the Cohen-Macauley property, Annals of Combinatorics, 10 (2006), 443-462.
• G. Hetyei, Central Delannoy numbers and balanced Cohen-Macaulay complexes, Ann. Comb. 10 (2006), 443-462.
• G. Hetyei, Links we almost missed between Delannoy numbers and Legendre polynomials
• Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
• M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• Clark Kimberling, Enumeration of paths, compositions of integers and Fibonacci numbers, Fib. Quarterly 39 (5) (2001) 430-435, Figure 1.
• Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3C.
• Benjamin Lyons and McCabe Olsen, Self-Reachable Chip Configurations on Trees, arXiv:2409.00763 [math.CO], 2024. See p. 18.
• Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
• Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Crossrefs

Diagonal sums are even-indexed Fibonacci numbers. Alternating (+-) diagonal sums are signed Fibonacci numbers.

T(n, n-1) = A001850(n) (Delannoy numbers). T(n, n)=A047781. Cf. A035028, A055587.

Cf. A208341. A055587.

Programs

Haskell

a049600 n k = a049600_tabl !! n !! k
a049600_row n = a049600_tabl !! n
a049600_tabl = [0] : map (0 :) a208341_tabl
-- Reinhard Zumkeller, Apr 15 2014

Maple

A049600 := proc(n,k)
    add(binomial(k+j,j)*binomial(n-1,j),j=0..n-1) ;
end proc: # R. J. Mathar, Oct 26 2015

Mathematica

t[n_, k_] := Hypergeometric2F1[ n-k+1, 1-k, 1, -1] // Floor; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2013 *)
t[n_, k_] := Sum[LaguerreL[n-k, i, 0]* LaguerreL[k-i, i, 0], {i,0,k}] //Floor; Table[t[n,k], {n, 0, 16}, {k, -1, n}] (* Richard Turk, Sep 08 2017 *)
T[n_, k_] := If[k == 0, 0, JacobiP[k - 1, 0, 1 - 2*k + n, 3]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 25 2021 *)

PARI

{A(i, j) = polcoeff( (x / (1 - 2*x)) * ((1 - x) / (1 - 2*x))^j + x * O(x^i), i)}; /* Michael Somos, Oct 01 2003 */

PARI

T(n,k)=sum(j=0,n-1,binomial(k+j,j)*binomial(n-1,j)) \\ Paul D. Hanna, Oct 06 2006

Formula

T(n,k) = Sum_{j=0..n-1} C(k+j,j)*C(n-1,j). - Paul D. Hanna, Oct 06 2006

T(i,j) = 2*T(i-1,j) + T(i,j-1) - T(i-1,j-1) with T(0,0)=1 and T(i,j)=0 if one of i,j<0. - Theodore Kolokolnikov, Jul 05 2010

O.g.f.: t*x/(1 - (2*t+1)*x + t*x^2) = t*x + (t + 2*t^2)*x^2 + (t + 3*t^2 + 4*t^3)*x^3 + .... Taking the row reverse of this triangle (with an additional column of 1's) gives A055587. - Peter Bala, Sep 10 2012

T(i,0) = 2^(i-1) and for j>0, T(i,j) = T(i,j-1) + Sum_{k=0..i-1} T(k,j). - Glen Whitney, Aug 17 2021

T(n, k) = JacobiP(k - 1, 0, 1 - 2*k + n, 3) for k >= 1. - Peter Luschny, Nov 25 2021