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A154380 The Riordan square of the Bell numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

%I A154380 #22 Jun 15 2019 07:24:47
%S A154380 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 A154380 11,1,877,1641,1235,683,277,77,13,1,4140,7451,5324,2987,1329,445,104,
%U A154380 15,1,21147,36525,24329,13391,6230,2340,669,135,17,1
%N A154380 The Riordan square of the Bell numbers. Triangle T(n, k), 0 <= k <= n, read by rows.
%C A154380 The Riordan square is defined in A321620.
%C A154380 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 A154380 In general, the triangle [r_0, r_1, r_2, ...] DELTA [s_0, s_1, s_2, ...] has generating function
%C A154380 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 A154380 A130167*A007318 as infinite lower triangular matrices. - _Philippe Deléham_, Jan 11 2009
%H A154380 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 A154380 G.f.: 1/(1-(x+xy)/(1-x/(1-x/(1-2x/(1-x/(1-3x/(1-x/(1-4x/(1-... (continued fraction).
%e A154380 Triangle begins
%e A154380      1;
%e A154380      1,   1;
%e A154380      2,   3,   1;
%e A154380      5,   9,   5,   1;
%e A154380     15,  29,  20,   7,  1;
%e A154380     52, 102,  77,  35,  9,  1;
%e A154380    203, 392, 302, 157, 54, 11, 1;
%p A154380 # The function RiordanSquare is defined in A321620.
%p A154380 RiordanSquare(add(x^k/mul(1-j*x, j=1..k), k=0..10), 10); # _Peter Luschny_, Dec 06 2018
%t A154380 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 A154380 RiordanSquare[Sum[x^k/Product[1 - j x, {j, 1, k}], {k, 0, 10}], 10] (* _Jean-François Alcover_, Jun 15 2019, from Maple *)
%Y A154380 First column are the Bell numbers A000110.
%Y A154380 Row sums are A154381, alternating row sums are A000007.
%Y A154380 Cf. A321620.
%K A154380 easy,nonn,tabl
%O A154380 0,4
%A A154380 _Paul Barry_, Jan 08 2009
%E A154380 New name by _Peter Luschny_, Dec 06 2018