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%I #22 Jun 15 2019 07:24:47
%S 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,
%T 11,1,877,1641,1235,683,277,77,13,1,4140,7451,5324,2987,1329,445,104,
%U 15,1,21147,36525,24329,13391,6230,2340,669,135,17,1
%N The Riordan square of the Bell numbers. Triangle T(n, k), 0 <= k <= n, read by rows.
%C The Riordan square is defined in A321620.
%C 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.
%C In general, the triangle [r_0, r_1, r_2, ...] DELTA [s_0, s_1, s_2, ...] has generating function
%C 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)
%C A130167*A007318 as infinite lower triangular matrices. - _Philippe Deléham_, Jan 11 2009
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6.
%F G.f.: 1/(1-(x+xy)/(1-x/(1-x/(1-2x/(1-x/(1-3x/(1-x/(1-4x/(1-... (continued fraction).
%e Triangle begins
%e 1;
%e 1, 1;
%e 2, 3, 1;
%e 5, 9, 5, 1;
%e 15, 29, 20, 7, 1;
%e 52, 102, 77, 35, 9, 1;
%e 203, 392, 302, 157, 54, 11, 1;
%p # The function RiordanSquare is defined in A321620.
%p RiordanSquare(add(x^k/mul(1-j*x, j=1..k), k=0..10), 10); # _Peter Luschny_, Dec 06 2018
%t 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}]];
%t RiordanSquare[Sum[x^k/Product[1 - j x, {j, 1, k}], {k, 0, 10}], 10] (* _Jean-François Alcover_, Jun 15 2019, from Maple *)
%Y First column are the Bell numbers A000110.
%Y Row sums are A154381, alternating row sums are A000007.
%Y Cf. A321620.
%K easy,nonn,tabl
%O 0,4
%A _Paul Barry_, Jan 08 2009
%E New name by _Peter Luschny_, Dec 06 2018
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