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A171488 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A005773(n+1)= 1,2,5,13,35,96,267,...

%I A171488 #18 Sep 30 2020 05:00:47
%S A171488 1,2,1,5,4,1,13,14,6,1,35,46,27,8,1,96,147,107,44,10,1,267,462,396,
%T A171488 204,65,12,1,750,1437,1404,858,345,90,14,1,2123,4438,4835,3388,1625,
%U A171488 538,119,16,1,6046,13637,16305,12802,7072,2805,791,152,18,1
%N A171488 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A005773(n+1)= 1,2,5,13,35,96,267,...
%C A171488 Equal to A064189*B = B*A054336 = B^(-1)*A035324, B = A007318.
%F A171488 Sum_{k, 0<=k<=n} T(n,k)*x^k = A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -2, -1, 0, 1 respectively.
%F A171488 T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*(-1)^i. - _Philippe Deléham_, Feb 23 2012
%F A171488 T(n,k) = (k+1)*Sum_{j=0..n-k} C(2*j+k,j)*(-1)^j*3^(n-j-k)*C(n+1,j+k+1)/(n+1). - _Vladimir Kruchinin_ Sep 30 2020
%e A171488 Triangle T(n,k) (0<=k<=n) begins:
%e A171488    1;
%e A171488    2,   1;
%e A171488    5,   4,   1;
%e A171488   13,  14,   6,  1;
%e A171488   35,  46,  27,  8,  1;
%e A171488   96, 147, 107, 44, 10, 1;
%e A171488   ...
%o A171488 (Maxima)
%o A171488 T(n,k)=((k+1)*sum(binomial(2*j+k,j)*(-1)^j*3^(n-j-k)*binomial(n+1,j+k+1),j,0,n-k))/(n+1); /* _Vladimir Kruchinin_ Sep 30 2020 */
%Y A171488 Cf. A097609, A064189.
%K A171488 nonn,tabl
%O A171488 0,2
%A A171488 _Philippe Deléham_, Dec 10 2009