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A154380
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The Riordan square of the Bell numbers. Triangle T(n, k), 0 <= k <= n, read by rows.
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3
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1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 15, 29, 20, 7, 1, 52, 102, 77, 35, 9, 1, 203, 392, 302, 157, 54, 11, 1, 877, 1641, 1235, 683, 277, 77, 13, 1, 4140, 7451, 5324, 2987, 1329, 445, 104, 15, 1, 21147, 36525, 24329, 13391, 6230, 2340, 669, 135, 17, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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The Riordan square is defined in A321620.
Previous name was: Triangle T(n,k), 0<=k<=n, read by rows given by [1, 1, 1, 2, 1, 3, 1, 4, 1, ...] DELTA [1, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
In general, the triangle [r_0, r_1, r_2, ...] DELTA [s_0, s_1, s_2, ...] has generating function
1/(1 - (r_0*x + s_0*x*y)/(1 - (r_1*x + s_1*x*y)/(1 - (r_2*x + s_2*x*y)/(1 -... (continued fraction)
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LINKS
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FORMULA
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G.f.: 1/(1-(x+xy)/(1-x/(1-x/(1-2x/(1-x/(1-3x/(1-x/(1-4x/(1-... (continued fraction).
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EXAMPLE
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Triangle begins
1;
1, 1;
2, 3, 1;
5, 9, 5, 1;
15, 29, 20, 7, 1;
52, 102, 77, 35, 9, 1;
203, 392, 302, 157, 54, 11, 1;
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MAPLE
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# The function RiordanSquare is defined in A321620.
RiordanSquare(add(x^k/mul(1-j*x, j=1..k), k=0..10), 10); # Peter Luschny, Dec 06 2018
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MATHEMATICA
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RiordanSquare[gf_, len_] := Module[{T}, T[n_, k_] := T[n, k] = If[k == 0, SeriesCoefficient[gf, {x, 0, n}], Sum[T[j, k - 1] T[n - j, 0], {j, k - 1, n - 1}]]; Table[T[n, k], {n, 0, len - 1}, {k, 0, n}]];
RiordanSquare[Sum[x^k/Product[1 - j x, {j, 1, k}], {k, 0, 10}], 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)
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CROSSREFS
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First column are the Bell numbers A000110.
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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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