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A213568
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Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
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8
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1, 4, 2, 11, 7, 3, 26, 18, 10, 4, 57, 41, 25, 13, 5, 120, 88, 56, 32, 16, 6, 247, 183, 119, 71, 39, 19, 7, 502, 374, 246, 150, 86, 46, 22, 8, 1013, 757, 501, 309, 181, 101, 53, 25, 9, 2036, 1524, 1012, 628, 372, 212, 116, 60, 28, 10, 4083, 3059, 2035, 1267
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OFFSET
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1,2
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COMMENTS
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Row 1, (1,3,6,...)**(1,4,9,...): A125128
Row 2, (1,3,6,...)**(4,9,16,...): A095151
Row 3, (1,3,6,...)**(9,16,25,...): A000247
Row 4, (1,3,6,...)**(16,25,36...): A208638 (?)
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) - 5*T(n,k-2) + 2*T(n,k-3). - corrected by Clark Kimberling, Sep 03 2023
G.f. for row n: f(x)/g(x), where f(x) = n - (n - 1)*x and g(x) = (1 - 2*x)*(1 - x)^2.
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EXAMPLE
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Northwest corner (the array is read by falling antidiagonals):
1...4....11...26....57....120
2...7....18...41....88....183
3...10...25...56....119...246
4...13...32...71....150...309
5...16...39...86....181...372
6...19...46...101...212...435
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MATHEMATICA
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(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n;
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}] (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Second program *)
Table[2^(n-k+1)*(k+1) -(n+2), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 26 2019 *)
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PROG
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(PARI) for(n=1, 12, for(k=1, n, print1(2^(n-k+1)*(k+1) -(n+2), ", "))) \\ G. C. Greubel, Jul 26 2019
(Magma) [2^(n-k+1)*(k+1) -(n+2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 26 2019
(Sage) [[2^(n-k+1)*(k+1) -(n+2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 26 2019
(GAP) Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*(k+1) -(n+2) ))); # G. C. Greubel, Jul 26 2019
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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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