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A213750
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Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 2*(n-1+h)-1, n>=1, h>=1, and ** = convolution.
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5
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1, 5, 3, 14, 11, 5, 30, 26, 17, 7, 55, 50, 38, 23, 9, 91, 85, 70, 50, 29, 11, 140, 133, 115, 90, 62, 35, 13, 204, 196, 175, 145, 110, 74, 41, 15, 285, 276, 252, 217, 175, 130, 86, 47, 17, 385, 375, 348, 308, 259, 205, 150, 98, 53, 19, 506, 495, 465, 420
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OFFSET
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1,2
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COMMENTS
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row 1, (1,2,3,4,5,...)**(1,3,5,7,9,...): A000330
row 2, (1,2,3,4,5,...)**(3,5,7,9,...): A051925
row 3, (1,2,3,4,5,...)**(5,7,9,11,...): (2*k^3 + 15*k^2 + 13*k)/6
row 4, (1,2,3,4,5,...)**(7,9,11,13,...): (2*k^3 + 21*k^2 + 19*k)/6
For a guide to related arrays, see A213500.
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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) = (2*n - 1) - (2*n - 3)*x and g(x) = (1 - x )^4.
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EXAMPLE
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Northwest corner (the array is read by falling antidiagonals):
1....5....14...30....55....91
3....11...26...50....85....133
5....17...38...70....115...175
7....23...50...90....145...217
9....29...62...110...175...259
11...35...74...130...205...301
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MATHEMATICA
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b[n_] := n; c[n_] := 2 n - 1;
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}] (* A213750 *)
d = Table[t[n, n], {n, 1, 40}] (* A007585 *)
s1 = Table[s[n], {n, 1, 50}] (* A002417 *)
FindLinearRecurrence[s1]
FindGeneratingFunction[s1, x]
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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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