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A213234
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Triangle read by rows: coefficients of auxiliary Rudin-Shapiro polynomials A_{ns}(omega) written in descending powers of x.
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4
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2, 1, 1, -2, 1, -3, 1, -4, 2, 1, -5, 5, 1, -6, 9, -2, 1, -7, 14, -7, 1, -8, 20, -16, 2, 1, -9, 27, -30, 9, 1, -10, 35, -50, 25, -2, 1, -11, 44, -77, 55, -11, 1, -12, 54, -112, 105, -36, 2, 1, -13, 65, -156, 182, -91, 13, 1, -14, 77, -210, 294, -196, 49, -2, 1, -15, 90, -275, 450, -378, 140, -15
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OFFSET
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0,1
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LINKS
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FORMULA
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EXAMPLE
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The first few polynomials are:
2
x
x^2-2
x^3-3*x
x^4+2-4*x^2
x^5-5*x^3+5*x
x^6-2-6*x^4+9*x^2
x^7-7*x^5+14*x^3-7*x
x^8+2-8*x^6+20*x^4-16*x^2
x^9-9*x^7+27*x^5-30*x^3+9*x
x^10-2-10*x^8+35*x^6-50*x^4+25*x^2
x^11-11*x^9+44*x^7-77*x^5+55*x^3-11*x
x^12+2-12*x^10+54*x^8-112*x^6+105*x^4-36*x^2
...
Triangle begins:
[2]
[1]
[1, -2]
[1, -3]
[1, -4, 2]
[1, -5, 5]
[1, -6, 9, -2]
[1, -7, 14, -7]
[1, -8, 20, -16, 2]
[1, -9, 27, -30, 9]
[1, -10, 35, -50, 25, -2]
[1, -11, 44, -77, 55, -11]
[1, -12, 54, -112, 105, -36, 2]
...
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MAPLE
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#The program is valid for n>=1:
f:=n->x^n+add((-1)^i*(n/i)*binomial(n-i-1, i-1)*x^(n-2*i), i=1..floor(n/2));
g:=n->series(x^n*subs(x=1/x, f(n)), x, n+1);
h:=n->seriestolist(series(subs(x=sqrt(x), g(n)), x, n+1));
for n from 0 to 15 do lprint(h(n)); od:
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MATHEMATICA
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Block[{t}, t[0, 0] = 2; t[n_, k_] := Binomial[n - k, k] + Binomial[n - k - 1, k - 1]; Table[(-1)^k*t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] ] // Flatten (* Michael De Vlieger, Jun 26 2020, after Jean-François Alcover at A034807 *)
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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