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%I #125 Feb 06 2024 05:12:01
%S 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,
%T 26,63,104,112,64,0,1,8,34,96,192,272,256,128,0,1,9,43,138,321,552,
%U 688,576,256,0,1,10,53,190,501,1002,1520,1696,1280,512
%N 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.
%C Essentially array A059576 divided by sequence A011782.
%C [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
%C Essentially triangle in A208341. - _Philippe Deléham_, Mar 23 2012
%C 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
%C 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
%H Reinhard Zumkeller, <a href="/A049600/b049600.txt">Rows n = 0..125 of table, flattened</a>
%H David Callan, <a href="https://arxiv.org/abs/2112.05241">Some bijections for lattice paths</a>, arXiv:2112.05241 [math.CO], 2021.
%H David Callan, <a href="https://arxiv.org/abs/2202.04649">A bijection for Delannoy paths</a>, arXiv:2202.04649 [math.CO], 2022.
%H R. Cori and G. Hetyei, <a href="http://arxiv.org/abs/1306.4628">Counting genus one partitions and permutations</a>, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
%H R. Cori and G. Hetyei, <a href="https://doi.org/10.46298/dmtcs.2404">How to count genus one partitions</a>, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
%H Robert Cori and Gabor Hetyei, <a href="https://hal.archives-ouvertes.fr/hal-01207612">Genus one partitions</a>, 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, <hal-01207612>.
%H Sergio Falcon, <a href="http://dx.doi.org/10.1080/23311835.2016.1201944">On the complex k-Fibonacci numbers, Cogent Mathematics, (2016), 3: 1201944. See Table 1.
%H G. Hetyei, <a href="http://www.math.uncc.edu/preprint/2005/2005_02.pdf">Central Delannoy numbers, Legendre polynomials and a balanced join operation preserving the Cohen-Macauley property</a>, Annals of Combinatorics, 10 (2006), 443-462.
%H G. Hetyei, <a href="https://doi.org/10.1007/s00026-006-0299-1">Central Delannoy numbers and balanced Cohen-Macaulay complexes</a>, Ann. Comb. 10 (2006), 443-462.
%H G. Hetyei, <a href="http://www.math.cornell.edu/event/conf/billera65/notes/hetyei.pdf">Links we almost missed between Delannoy numbers and Legendre polynomials</a>
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H M. Janjic and B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H Clark Kimberling, <a href="https://www.fq.math.ca/Scanned/39-5/kimberling.pdf">Enumeration of paths, compositions of integers and Fibonacci numbers</a>, Fib. Quarterly 39 (5) (2001) 430-435, Figure 1.
%H Clark Kimberling, <a href="https://www.fq.math.ca/Scanned/40-4/kimberling.pdf">Path-counting and Fibonacci numbers</a>, Fib. Quart. 40 (4) (2002) 328-338, Example 3C.
%H Thomas Selig, <a href="https://arxiv.org/abs/2202.06487">Combinatorial aspects of sandpile models on wheel and fan graphs</a>, arXiv:2202.06487 [math.CO], 2022.
%H Luis Verde-Star <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
%F T(n,k) = Sum_{j=0..n-1} C(k+j,j)*C(n-1,j). - _Paul D. Hanna_, Oct 06 2006
%F 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
%F 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
%F 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
%F T(n, k) = JacobiP(k - 1, 0, 1 - 2*k + n, 3) for k >= 1. - _Peter Luschny_, Nov 25 2021
%e Diagonals (each starting on row 1): {0}; {0,1}; {0,1,2}; ...
%e Array begins:
%e 0 0 0 0 0 0 0 0 0 0 0 ...
%e 1 1 1 1 1 1 1 1 1 1 1 ...
%e 2 3 4 5 6 7 8 9 10 11 12 ...
%e 4 8 13 19 26 34 43 53 64 76 89 ...
%e 8 20 38 63 96 138 190 253 328 416 518 ...
%e 16 48 104 192 321 501 743 1059 1462 1966 2586 ...
%e 32 112 272 552 1002 1683 2668 4043 5908 8378 11584 ...
%e 64 256 688 1520 2972 5336 8989 14407 22180 33028 47818 ...
%e Triangle begins:
%e 0;
%e 0, 1;
%e 0, 1, 2;
%e 0, 1, 3, 4;
%e 0, 1, 4, 8, 8;
%e 0, 1, 5, 13, 20, 16;
%e 0, 1, 6, 19, 38, 48, 32;
%e 0, 1, 7, 26, 63, 104, 112, 64;
%e ...
%e (1, 0, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) where DELTA is the operator defined in A084938 begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 3, 4, 0;
%e 1, 4, 8, 8, 0;
%e 1, 5, 13, 20, 16, 0;
%e 1, 6, 19, 38, 48, 32, 0;
%e 1, 7, 26, 63, 104, 112, 64, 0;
%p A049600 := proc(n,k)
%p add(binomial(k+j,j)*binomial(n-1,j),j=0..n-1) ;
%p end proc: # _R. J. Mathar_, Oct 26 2015
%t 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 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 T[n_, k_] := If[k == 0, 0, JacobiP[k - 1, 0, 1 - 2*k + n, 3]];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Peter Luschny_, Nov 25 2021 *)
%o (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 */
%o (PARI) T(n,k)=sum(j=0,n-1,binomial(k+j,j)*binomial(n-1,j)) \\ _Paul D. Hanna_, Oct 06 2006
%o (Haskell)
%o a049600 n k = a049600_tabl !! n !! k
%o a049600_row n = a049600_tabl !! n
%o a049600_tabl = [0] : map (0 :) a208341_tabl
%o -- _Reinhard Zumkeller_, Apr 15 2014
%Y Diagonal sums are even-indexed Fibonacci numbers. Alternating (+-) diagonal sums are signed Fibonacci numbers.
%Y T(n, n-1) = A001850(n) (Delannoy numbers). T(n, n)=A047781. Cf. A035028, A055587.
%Y Cf. A208341. A055587.
%K nonn,tabl,easy
%O 0,6
%A _Clark Kimberling_
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