A171488 - OEIS

%I #18 Sep 30 2020 05:00:47

%S 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 204,65,12,1,750,1437,1404,858,345,90,14,1,2123,4438,4835,3388,1625,

%U 538,119,16,1,6046,13637,16305,12802,7072,2805,791,152,18,1

%N 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 Equal to A064189*B = B*A054336 = B^(-1)*A035324, B = A007318.

%F 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 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 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 Triangle T(n,k) (0<=k<=n) begins:

%e    1;

%e    2,   1;

%e    5,   4,   1;

%e   13,  14,   6,  1;

%e   35,  46,  27,  8,  1;

%e   96, 147, 107, 44, 10, 1;

%e   ...

%o (Maxima)

%o 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 Cf. A097609, A064189.

%K nonn,tabl

%O 0,2

%A _Philippe Deléham_, Dec 10 2009