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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)= 0, j >= 0. 27
 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, 0, 1, 11, 64, 253, 743 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 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 REFERENCES Robert Cori, 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, C. Kimberling, Enumeration of paths, compositions of integers, and Fibonacci numbers, Fib. Q. 39 (5) (2001) 430-435, Figure 1. C. Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3C. LINKS Reinhard Zumkeller, Rows n = 0..125 of table, flattened R. Cori, G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013. R. Cori, G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344. 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. M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013 M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5 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) = 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] 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 EXAMPLE 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 . 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 *) PROG (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 (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 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. Sequence in context: A263097 A286011 A241954 * A318602 A004542 A207331 Adjacent sequences:  A049597 A049598 A049599 * A049601 A049602 A049603 KEYWORD nonn,tabl,easy AUTHOR STATUS approved

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Last modified January 18 09:00 EST 2019. Contains 319269 sequences. (Running on oeis4.)