A109267 - OEIS

%I #30 Mar 15 2020 02:29:50

%S 1,1,1,3,2,1,9,6,3,1,29,19,10,4,1,97,63,34,15,5,1,333,215,118,55,21,6,

%T 1,1165,749,416,201,83,28,7,1,4135,2650,1485,736,320,119,36,8,1,14845,

%U 9490,5355,2705,1220,484,164,45,9,1,53791,34318,19473,9983,4628,1923,703,219,55,10,1

%N Riordan array (1/(1 - x*c(x) - x^2*c(x)^2), x*c(x)) where c(x) is the g.f. of A000108.

%C Inverse of Riordan array (1-x-x^2, x(1-x)), A109264. Row sums are A109262(n+1). Diagonal sums are A109268. Columns include A081696, A109262, A109263.

%H Muniru A Asiru, <a href="/A109267/b109267.txt">Table of n, a(n) for n = 0..5150</a>

%H Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.

%F The production matrix M (deleting the zeros) is:

%F   1, 1;

%F   2, 1, 1;

%F   2, 1, 1, 1;

%F   2, 1, 1, 1, 1;

%F ... such that the n-th row of the triangle is the top row of M^n. - _Gary W. Adamson_, Feb 16 2012

%F From _Peter Bala_, Feb 18 2018: (Start)

%F T(n,k) = Sum_{i = 0..n-k} (Fibonacci(i+1) - 2*Fibonacci(i))* binomial(2*n-k-i,n), 0 <= k <= n.

%F The n-th row polynomial of the row reverse triangle equals the n-th degree Taylor polynomial of the function (1 - 2*x)/((1 - x)*(1 - x - x^2)) * 1/(1 - x)^n about 0. For example, for n = 4, (1 - 2*x)/((1 - x)*(1 - x - x^2)) * 1/(1 - x)^4 = 1 + 4*x + 10*x^2 + 19*x^3 + 29*x^4 + O(x^5), giving (29, 19, 10, 4, 1) as row 4. (End)

%e Rows begin

%e    1;

%e    1,  1;

%e    3,  2,  1;

%e    9,  6,  3,  1;

%e   29, 19, 10,  4,  1;

%e   97, 63, 34, 15,  5,  1;

%p A109267 := (n,k) -> add(-combinat:-fibonacci(i-2)*binomial(2*n-k-i,n), i=0..n-k):

%p seq(seq(A109267(n, k), k = 0..n), n = 0..10); # _Peter Bala_, Feb 18 2018

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

%t c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);

%t RiordanArray[1/(1 - # c[#] - #^2 c[#]^2)&, # c[#]&, 11] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)

%o (GAP) Flat(List([0..10],n->List([0..n],k->Sum([0..n-k],i->(Fibonacci(i+1)-2*Fibonacci(i))*Binomial(2*n-k-i,n))))); # _Muniru A Asiru_, Feb 19 2018

%Y Row sums A109262, sums along shallow diagonals A109268, A081696 (column 0), A109262 (column 1), A109263 (column 2).

%Y Cf. A000108, A109264.

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Jun 24 2005