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A171568 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A064613.

%I A171568 #28 Feb 23 2024 08:01:13
%S A171568 1,3,1,10,6,1,37,29,9,1,150,134,57,12,1,654,622,318,94,15,1,3012,2948,
%T A171568 1686,616,140,18,1,14445,14317,8781,3693,1055,195,21,1,71398,71142,
%U A171568 45625,21132,7075,1662,259,24,1,361114,360602,238170,118042,44303,12345,2464,332,27,1
%N A171568 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A064613.
%C A171568 Equal to A171515*B = B*A104259, B = A007318.
%F A171568 T(n, 0) - T(n, 1) = 2^n.
%F A171568 T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + Sum_{i=0..n} T(n-1, k+1+i). - _Philippe Deléham_, Feb 23 2012
%e A171568 Triangle T(n,k) begins
%e A171568 [0]     1;
%e A171568 [1]     3,     1;
%e A171568 [2]    10,     6,     1;
%e A171568 [3]    37,    29,     9,     1;
%e A171568 [4]   150,   134,    57,    12,    1;
%e A171568 [5]   654,   622,   318,    94,   15,    1;
%e A171568 [6]  3012,  2948,  1686,   616,  140,   18,   1;
%e A171568 [7] 14445, 14317,  8781,  3693, 1055,  195,  21,  1;
%e A171568 [8] 71398, 71142, 45625, 21132, 7075, 1662, 259, 24, 1;
%e A171568 .
%e A171568 Production array begins
%e A171568   3, 1
%e A171568   1, 3, 1
%e A171568   1, 1, 3, 1
%e A171568   1, 1, 1, 3, 1
%e A171568   1, 1, 1, 1, 3, 1
%e A171568   1, 1, 1, 1, 1, 3, 1
%e A171568 - _Philippe Deléham_, Mar 05 2013
%p A171568 T := proc(n,k) option remember;
%p A171568 if n < 0 or k < 0 then 0 elif n = k then 1 else
%p A171568 T(n-1, k-1) + 3*T(n-1,k) + add(T(n-1, k+1+i), i=0..n) fi end:
%p A171568 for n from 0 to 8 do seq(T(n,k), k = 0..n) od; # _Peter Luschny_, Oct 16 2022
%t A171568 T[n_, k_] := T[n, k] = If[n < 0 || k < 0, 0, If[n == k, 1, T[n-1, k-1] + 3*T[n-1, k] + Sum[T[n-1, k+1+i], {i, 0, n}]]];
%t A171568 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 23 2024, after _Peter Luschny_ *)
%Y A171568 Sum_{k=0..n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x = -1, 0, 1, 2 respectively.
%Y A171568 Cf. A007318, A052179, A171589, A171515, A104259.
%K A171568 nonn,tabl
%O A171568 0,2
%A A171568 _Philippe Deléham_, Dec 11 2009
%E A171568 Corrected and extended by _Peter Luschny_, Oct 16 2022