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A049213
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A convolution triangle of numbers obtained from A025749.
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4
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1, 6, 1, 56, 12, 1, 616, 148, 18, 1, 7392, 1904, 276, 24, 1, 93632, 25312, 4080, 440, 30, 1, 1230592, 344960, 59808, 7360, 640, 36, 1, 16612992, 4792128, 876960, 118224, 11960, 876, 42, 1, 228890112, 67586816, 12900416, 1860992, 209200, 18096, 1148
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OFFSET
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1,2
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COMMENTS
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G.f. for m-th column: ((1-(1-16*x)^(1/4))/4)^m.
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LINKS
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FORMULA
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a(n, m) = 4*(4*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0; a(1, 1)=1.
a(n,m) = (m/n) * 4^(n-m) * Sum_{k=1..n-m} binomial(n+k-1, n-1) * Sum_{j=0..k} binomial(j, n-m-3*k+2*j) * 4^(j-k) * binomial(k,j) * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j), n > m; a(n,n)=1. - Vladimir Kruchinin, Feb 08 2011
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MATHEMATICA
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a[n_, n_] = 1; a[n_, m_] := m/n * 4^(n-m) * Sum[ Binomial[n+k-1, n-1] * Sum[ Binomial[j, n-m-3*k+2*j] * 4^(j-k) * Binomial[k, j] * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j), {j, 0, k}], {k, 1, n-m}]; Table[a[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
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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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