A109267 Riordan array (1/(1 - x*c(x) - x^2*c(x)^2), x*c(x)) where c(x) is the g.f. of A000108.
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, 1, 1165, 749, 416, 201, 83, 28, 7, 1, 4135, 2650, 1485, 736, 320, 119, 36, 8, 1, 14845, 9490, 5355, 2705, 1220, 484, 164, 45, 9, 1, 53791, 34318, 19473, 9983, 4628, 1923, 703, 219, 55, 10, 1
Offset: 0
Examples
Rows begin 1; 1, 1; 3, 2, 1; 9, 6, 3, 1; 29, 19, 10, 4, 1; 97, 63, 34, 15, 5, 1;
Links
- Muniru A Asiru, Table of n, a(n) for n = 0..5150
- Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Crossrefs
Programs
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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
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Maple
A109267 := (n,k) -> add(-combinat:-fibonacci(i-2)*binomial(2*n-k-i,n), i=0..n-k): seq(seq(A109267(n, k), k = 0..n), n = 0..10); # Peter Bala, Feb 18 2018
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Mathematica
(* The function RiordanArray is defined in A256893. *) c[x_] := (1 - Sqrt[1 - 4 x])/(2 x); RiordanArray[1/(1 - # c[#] - #^2 c[#]^2)&, # c[#]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
Formula
The production matrix M (deleting the zeros) is:
1, 1;
2, 1, 1;
2, 1, 1, 1;
2, 1, 1, 1, 1;
... such that the n-th row of the triangle is the top row of M^n. - Gary W. Adamson, Feb 16 2012
From Peter Bala, Feb 18 2018: (Start)
T(n,k) = Sum_{i = 0..n-k} (Fibonacci(i+1) - 2*Fibonacci(i))* binomial(2*n-k-i,n), 0 <= k <= n.
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)
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