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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 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. 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). Cf. A000108, A109264. Sequence in context: A160760 A152860 A002350 * A185416 A193918 A298804 Adjacent sequences: A109264 A109265 A109266 * A109268 A109269 A109270 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.)