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A144400
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Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)*binomial(n, j)*x^(j - 1)*(1 - x)^(n - j).
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4
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1, 2, -1, 3, -3, 1, 4, -6, 4, 0, 5, -10, 10, 0, -3, 6, -15, 20, 0, -18, 10, 7, -21, 35, 0, -63, 70, -24, 8, -28, 56, 0, -168, 280, -192, 49, 9, -36, 84, 0, -378, 840, -864, 441, -89, 10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145, 11, -55, 165, 0
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f.: (y - (1 - 2*x)*y^2)/(1 - 3*(1 - x)*y + (3 - 6*x + 2*x^2)*y^2 - (1 - 3*x + 2*x^2 + x^3)*y^3). - Franck Maminirina Ramaharo, Oct 22 2018
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EXAMPLE
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Triangle begins:
1;
2, -1;
3, -3, 1;
4, -6, 4, 0;
5, -10, 10, 0, -3;
6, -15, 20, 0, -18, 10;
7, -21, 35, 0, -63, 70, -24;
8, -28, 56, 0, -168, 280, -192, 49;
9, -36, 84, 0, -378, 840, -864, 441, -89;
10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145;
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MATHEMATICA
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a[n_]:= a[n]= If[n<3, Fibonacci[n], a[n-2] + a[n-3]];
p[x_, n_]:= Sum[a[k]*Binomial[n, k]*x^(k-1)*(1-x)^(n-k), {k, 0, n}];
Table[Coefficient[p[x, n], x, k], {n, 12}, {k, 0, n-1}]//Flatten
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PROG
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(Sage)
@CachedFunction
def f(n): return fibonacci(n) if (n<3) else f(n-2) + f(n-3)
def p(n, x): return sum( binomial(n, j)*f(j)*x^(j-1)*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
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CROSSREFS
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Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A174128.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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