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A118384
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Gaussian column reduction of Hankel matrix for central Delannoy numbers.
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5
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1, 3, 1, 13, 6, 1, 63, 33, 9, 1, 321, 180, 62, 12, 1, 1683, 985, 390, 100, 15, 1, 8989, 5418, 2355, 720, 147, 18, 1, 48639, 29953, 13923, 4809, 1197, 203, 21, 1, 265729, 166344, 81340, 30744, 8806, 1848, 268, 24, 1, 1462563, 927441, 471852, 191184, 60858
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OFFSET
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0,2
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COMMENTS
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First column is central Delannoy numbers A001850. Second column is A050151.
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LINKS
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FORMULA
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Number triangle T(n,k) = sum{j=0..n, C(n,j)C(j,n-k-j)2^(n-k-j)3^(2j-(n-k))}; Riordan array (1/sqrt(1-6x+x^2), (1-3x-sqrt(1-6x+x^2))/(4x)); Column k has e.g.f. exp(3x)Bessel_I(k,2*sqrt(2)x)/(sqrt(2))^k.
a(n) = sum(binomial(n,i)*binomial(n,n-k-i)*2^i,i=0..n), also a(n+1,k+1) = a(n,k) + 3*a(n,k+1) + 2*a(n,k+2). - Emanuele Munarini, Mar 16 2011
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - 3*x - sqrt(1 - 6*x + x^2) )/(4*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan, Jan 2000, Example 5.2).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + 3*x + 2*x^2. In general the (n,k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
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EXAMPLE
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Triangle begins
1,
3, 1,
13, 6, 1,
63, 33, 9, 1,
321, 180, 62, 12, 1,
1683, 985, 390, 100, 15, 1
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MATHEMATICA
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Table[Sum[Binomial[n, i]Binomial[n, n-k-i]2^i, {i, 0, n-k}], {n, 0, 8}, {k, 0, 8}]//MatrixForm
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PROG
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(Maxima) create_list(sum(binomial(n, i)*binomial(n, n-k-i)*2^i, i, 0, n), n, 0, 8, k, 0, n);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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