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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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.
LINKS
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 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)
EXAMPLE
Rows begin
1;
1, 1;
3, 2, 1;
9, 6, 3, 1;
29, 19, 10, 4, 1;
97, 63, 34, 15, 5, 1;
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
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 *)
PROG
(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
CROSSREFS
Row sums A109262, sums along shallow diagonals A109268, A081696 (column 0), A109262 (column 1), A109263 (column 2).
Sequence in context: A160760 A152860 A002350 * A185416 A193918 A298804
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Jun 24 2005
STATUS
approved

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Last modified April 19 07:38 EDT 2024. Contains 371782 sequences. (Running on oeis4.)