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A237596
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Convolution triangle of A000958(n+1).
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1
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1, 1, 1, 3, 2, 1, 8, 7, 3, 1, 24, 22, 12, 4, 1, 75, 73, 43, 18, 5, 1, 243, 246, 156, 72, 25, 6, 1, 808, 844, 564, 283, 110, 33, 7, 1, 2742, 2936, 2046, 1092, 465, 158, 42, 8, 1, 9458, 10334, 7449, 4178, 1906, 714, 217, 52, 9, 1, 33062, 36736, 27231, 15904, 7670, 3096, 1043, 288, 63, 10, 1
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OFFSET
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0,4
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COMMENTS
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Riordan array (f(x)/x, f(x)) where f(x) is the g.f. of A000958.
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LINKS
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FORMULA
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G.f.: for the column k-1: ((1-sqrt((1-4*x))^k/(1+sqrt(1-4*x) + 2*x)^k)/x.
Sum_{k=0..n} T(n,k) = A109262(n+1).
T(n, k) = coefficient of [x^k]( p(n+1, x) ), where p(n, x) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n-j)*Fibonacci(j, x) with p(0, x) = 1 and Fibonacci(n, x) are the Fibonacci polynomials.
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EXAMPLE
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Triangle begins:
1;
1, 1;
3, 2, 1;
8, 7, 3, 1;
24, 22, 12, 4, 1;
75, 73, 43, 18, 5, 1;
243, 246, 156, 72, 25, 6, 1;
808, 844, 564, 283, 110, 33, 7, 1;
...
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MAPLE
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# Uses function PMatrix from A357368. Adds column 1, 0, 0, 0, ... to the left.
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MATHEMATICA
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P[n_, x_]:= P[n, x]= If[n==0, 1, Sum[(j/(2*n-j))*Binomial[2*n-j, n-j]*Fibonacci[j, x], {j, 0, n}]];
T[n_, k_] := Coefficient[P[n+1, x], x, k];
Table[T[n, k], {n, 0, 13}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2022 *)
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PROG
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(SageMath)
def f(n, x): return sum( binomial(n-j-1, j)*x^(n-2*j-1) for j in (0..(n-1)//2) )
def p(n, x):
if (n==0): return 1
else: return sum( (j/(2*n-j))*binomial(2*n-j, n-j)*f(j, x) for j in (0..n) )
def A237596(n, k): return ( p(n+1, x) ).series(x, n+1).list()[k]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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