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A213849
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Rectangular array: (row n) = b**c, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and ** = convolution.
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4
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1, 2, 1, 5, 3, 2, 8, 6, 4, 2, 14, 11, 9, 5, 3, 20, 17, 14, 10, 6, 3, 30, 26, 23, 17, 13, 7, 4, 40, 36, 32, 26, 20, 14, 8, 4, 55, 50, 46, 38, 32, 23, 17, 9, 5, 70, 65, 60, 52, 44, 35, 26, 18, 10, 5, 91, 85, 80, 70, 62, 50, 41, 29, 21, 11, 6
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OFFSET
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1,2
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COMMENTS
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Row 1, (1,1,2,2,3,3,...)**(1,1,2,2,3,3,...).
Row 2, (1,1,2,2,3,3,...)**(1,2,2,3,3,4,...).
Row 3, (1,1,2,2,3,3,...)**(2,2,3,3,4,4,...).
For a guide to related arrays, see A212500.
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LINKS
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FORMULA
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T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(ceiling(n/2) + m(n)*x - floor(n/2)*x^2), where m(n) = (n+1 mod 2), and g(x) = (1+x)^2 *(1-x)^4.
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EXAMPLE
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Northwest corner (the array is read by falling antidiagonals):
1...2...5....8....14...20...30...40
1...3...6....11...17...26...36...50
2...4...9....14...23...32...46...60
2...5...10...17...26...38...52...70
3...6...13...20...32...44...62...80
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MATHEMATICA
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b[n_]:=Floor[(n+1)/2]; c[n_]:=Floor[(n+1)/2];
t[n_, k_]:=Sum[b[k-i]c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:=Table[t[n, k], {k, 1, 60}] (* A213849 *)
d=Table[t[n, n], {n, 1, 50}] (* A049778 *)
s[n_]:=Sum[t[i, n+1-i], {i, 1, n}]
s1=Table[s[n], {n, 1, 50}] (* A213850 *)
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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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