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A073714
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Table, read by antidiagonals, generated by successive convolutions of the first row such that the first row equals the same table in this flattened form.
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0
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 2, 1, 5, 10, 10, 7, 1, 1, 6, 15, 20, 18, 8, 1, 1, 7, 21, 35, 39, 27, 9, 3, 1, 8, 28, 56, 75, 68, 37, 14, 3, 1, 9, 36, 84, 132, 146, 108, 54, 20, 1, 1, 10, 45, 120, 217, 282, 260, 168, 81, 20, 1, 1, 11, 55, 165, 338, 504, 552, 440, 263, 106
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graph;
refs;
listen;
history;
text;
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OFFSET
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0,5
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LINKS
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FORMULA
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Let first row sequence be a(n)=T(0, n); define f(x) = sum_{k=0..inf} a(k)x^k, then the n-th row is generated by: f(x)^(n+1) = sum_{k=0..inf} T(n, k)x^k.
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EXAMPLE
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Table read by antidiagonals gives first row; subsequent rows generated by convolutions of first row sequence.
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4,...;
1, 2, 3, 4, 7, 8, 9, 14, 20, 20, 21, 32,...;
1, 3, 6, 10, 18, 27, 37, 54, 81, 106, 132, 180,...;
1, 4, 10, 20, 39, 68, 54, 168, 263, 388, 544, 768,...;
1, 5, 15, 35, 75, 108, 260, 440, 730,1165,1781,2670,...;
1, 6, 21, 56, 132, 282, 552,1014,1794,3058,5013,...;
1, 7, 28, 84, 217, 504,1071,2122,4004,7252,...;
1, 8, 36, 120, 338, 848,1940,4120,8271,...;
1, 9, 45, 165, 504,1359,3327,7533,...;
1,10, 55, 220, 725,2092,5455,...;
1,11, 66, 286,1012,3113,...;
1,12, 78, 364,1377,...;
1,13, 91, 455,...;
1,14,105,...;
1,15,...; ...
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MATHEMATICA
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max = 75; a[0] = 1; se[n_] := se[n] = Series[ Sum[x^(j*(j + 1)/2)*(1 + x)^j, {j, 0, max - n}]^(n + 1), {x, 0, max - n}]; t[n_, k_] := t[n, k] = Coefficient[se[n], x, k]; ft = Flatten[ Table[t[n - j, j], {n, 0, max}, {j, 0, n}]][[1 ;; max + 1]]; sol = Thread[ft == Table[a[k], {k, 0, max}]] // Solve; sol /. Rule -> Set; Table[a[k], {k, 0, max}] (* Jean-François Alcover, Aug 05 2013 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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