A110165 - OEIS

%I #22 Dec 29 2023 13:27:09

%S 1,3,1,11,6,1,45,30,9,1,195,144,58,12,1,873,685,330,95,15,1,3989,3258,

%T 1770,630,141,18,1,18483,15533,9198,3801,1071,196,21,1,86515,74280,

%U 46928,21672,7210,1680,260,24,1,408105,356283,236736,119154,44982,12510,2484,333,27,1

%N Riordan array (1/sqrt(1-6x+5x^2),(1-3x-sqrt(1-6x+5x^2))/(2x)).

%C Columns include A026375, A026376 and A026377. Inverse is A110168. Rows sums are A110166. Diagonal sums are A110167.

%C From _Peter Bala_, Jan 09 2022: (Start)

%C This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - 3*x - sqrt(1 - 6*x + 5*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).

%C T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + 3*x + x^2. In general the (n,k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

%H P. Peart and W.-J. Woan, <a href="https://doi.org/10.1016/S0166-218X(99)00166-3">A divisibility property for a subgroup of Riordan matrices</a>, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

%F Number triangle T(n, k) = Sum_{j = 0..n} C(n, j)C(2j, j+k).

%F T(n,0) = 3*T(n-1,0) + 2*T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k > 0, T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Jan 24 2014

%e Rows begin

%e     1;

%e     3,   1;

%e    11,   6,   1;

%e    45,  30,   9,   1;

%e   195, 144,  58,  12,   1;

%e   873, 685, 330,  95,  15,   1;

%e Production array begins:

%e   3, 1;

%e   2, 3, 1;

%e   0, 1, 3, 1;

%e   0, 0, 1, 3, 1;

%e   0, 0, 0, 1, 3, 1;

%e   0, 0, 0, 0, 1, 3, 1;

%e   0, 0, 0, 0, 0, 1, 3, 1;

%e   ... - _Philippe Deléham_, Feb 08 2014

%p seq(seq( coeff((x^2 + 3*x + 1)^n, x, n-k), k = 0..n ), n = 0..10); # _Peter Bala_, Jan 09 2022

%t (* The function RiordanArray is defined in A256893. *)

%t RiordanArray[1/Sqrt[1-6#+5#^2]&, (1-3#-Sqrt[1-6#+5#^2])/(2#)&, 10] // Flatten (* _Jean-François Alcover_, Jul 19 2019 *)

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Jul 14 2005