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A049600 Array T read by diagonals; T(i,j)=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. 27

%I

%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 688,576,256,0,1,10,53,190,501,1002,1520,1696,1280,512,0,1,11,64,253,743

%N Array T read by diagonals; T(i,j)=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

%H Reinhard Zumkeller, <a href="/A049600/b049600.txt">Rows n = 0..125 of table, flattened</a>

%H R. Cori, 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, G. Hetyei, <a href="http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/viewFile/dmAT0130/4488">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, 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 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, 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 C. 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 C. 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.

%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) = T(i-1,j)+2T(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

%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 (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 *)

%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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Last modified February 21 23:29 EST 2019. Contains 320381 sequences. (Running on oeis4.)