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A049600
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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.
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23
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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; internal format)
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OFFSET
| 0,6
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COMMENTS
| Essentially array A059576 divided by sequence A011782
[Hetyei] calls the transpose 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. [From Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008]
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. [From Theodore Kolokolnikov (tkolokol(AT)mathstat.dal.ca), Jul 05 2010]
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LINKS
| G. Hetyei, Central Delannoy numbers, Legendre polynomials and a balanced join operation preserving the Cohen-Macauley property, Annals of Combinatorics, 10 (2006), 443-462. [From Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008]
G. Hetyei, Links we almost missed between Delannoy numbers and Legendre polynomials [From Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008]
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FORMULA
| T(n,k) = Sum_{j=0..n-1} C(k+j,j)*C(n-1,j). - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 06 2006
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EXAMPLE
| Diagonals (each starting on row 1): {0}; {0,1}; {0,1,2}; ...
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PROG
| (PARI) A(i, j)=polcoeff(x/(1-2*x)*((1-x)/(1-2*x))^j+x*O(x^i), i) (from Michael Somos)
(PARI) T(n, k)=sum(j=0, n-1, binomial(k+j, j)*binomial(n-1, j)) - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 06 2006
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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.
Sequence in context: A068920 A099390 A124031 * A004542 A134405 A091612
Adjacent sequences: A049597 A049598 A049599 * A049601 A049602 A049603
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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