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A330795
Evaluation of the polynomials given by the Riordan square of the Fibonacci sequence with a(0) = 1 (A193737) at 1/2 and normalized with 2^n.
1
1, 3, 9, 39, 153, 615, 2457, 9831, 39321, 157287, 629145, 2516583, 10066329, 40265319, 161061273, 644245095, 2576980377, 10307921511, 41231686041, 164926744167, 659706976665, 2638827906663, 10555311626649, 42221246506599, 168884986026393, 675539944105575
OFFSET
0,2
FORMULA
a(n) = 2^n*Sum_{k=0..n} A193737(n,k)/2^k.
a(n) = [x^n] (1 - 4*x^2)/(1 - x*(3 + 4*x)).
a(n) = n! [x^n] (3/5)*exp(-x)*(exp(5*x) - 1) + 1.
a(n) = 4*a(n-2) + 3*a(n-1).
a(n) = 3*A015521(n), n>0. - R. J. Mathar, Aug 19 2022
MAPLE
gf := (4*x^2 - 1)/(x*(4*x + 3) - 1): ser := series(gf, x, 32):
seq(coeff(ser, x, n), n=0.. 25);
# Alternative:
gf:= (3/5)*exp(-x)*(exp(5*x) - 1) + 1: ser := series(gf, x, 32):
seq(n!*coeff(ser, x, n), n=0.. 25);
# Or:
a := proc(n) option remember; if n < 3 then return [1, 3, 9][n + 1] fi;
4*a(n-2) + 3*a(n-1) end: seq(a(n), n=0..25);
MATHEMATICA
LinearRecurrence[{3, 4}, {1, 3, 9}, 31] (* G. C. Greubel, Sep 14 2023 *)
PROG
(Magma) [1] cat [3*(4^n -(-1)^n)/5: n in [1..30]]; // G. C. Greubel, Sep 14 2023
(SageMath) [3*(4^n -(-1)^n)//5 + int(n==0) for n in range(31)] # G. C. Greubel, Sep 14 2023
CROSSREFS
Cf. A006131, A015521, A193737, A321620, A324969 (Fibonacci with a(0)=1).
Sequence in context: A296102 A149026 A149027 * A180741 A121101 A280066
KEYWORD
nonn,easy
AUTHOR
Peter Luschny, Jan 10 2020
STATUS
approved