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A157077
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Triangle read by rows, coefficients of the Legendre polynomials P(n, x) times 2^n: T(n, k) = 2^n * [x^k] P(n, x), n >= 0, 0 <= k <= n.
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0
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1, 0, 2, -2, 0, 6, 0, -12, 0, 20, 6, 0, -60, 0, 70, 0, 60, 0, -280, 0, 252, -20, 0, 420, 0, -1260, 0, 924, 0, -280, 0, 2520, 0, -5544, 0, 3432, 70, 0, -2520, 0, 13860, 0, -24024, 0, 12870, 0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620, -252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756
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OFFSET
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0,3
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LINKS
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FORMULA
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Row sums are 2^n.
T(n,1) = (-1)^floor(n/2)*A005430(floor(n/2)+1) if n is odd else 0.
Let Q(n, x) = 2^n*P(n, x).
Q(n,0) = (-1)^floor(n/2)*A126869(floor(n/2)) if n is even else 0.
n*T(n,k) = 2*(2*n-1)*T(n-1,k-1) - 4*(n-1)*T(n-2,k).
T(n,k) = (-1)^floor((n-k)/2)*binomial(n+k,k)*binomial(n,floor((n-k)/2))*(1+(-1)^(n-k))/2.
O.g.f.: A(x,t) = 1/sqrt(1-4*x*t+4*x^2) = 1 + (2*t)*x + (-2+6*t^2)*x^2 + (-12*t+20*t^3)*x^3 + (6-60*t^2+70*t^4)*x^4 + .... (End)
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EXAMPLE
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The term order is Q(x) = a_0 + a_1*x + ... + a_n*x^n. The coefficients of the first few polynomials in this order are:
{1},
{0, 2},
{-2, 0, 6},
{0, -12, 0, 20},
{6, 0, -60, 0, 70},
{0, 60, 0, -280, 0, 252},
{-20, 0, 420, 0, -1260, 0, 924},
{0, -280, 0, 2520, 0, -5544, 0, 3432},
{70, 0, -2520, 0, 13860, 0, -24024, 0, 12870},
{0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620},
{-252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756}.
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As a right-aligned triangle:
1;
0, 2;
-2, 0, 6;
0, -12, 0, 20;
6, 0, -60, 0, 70;
0, 60, 0, -280, 0, 252;
-20, 0, 420, 0, -1260, 0, 924;
0, -280, 0, 2520, 0, -5544, 0, 3432;
70, 0, -2520, 0, 13860, 0, -24024, 0, 12870;
0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620;
-252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756. (End)
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MAPLE
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with(orthopoly):with(PolynomialTools): seq(print(CoefficientList (2^n*P(n, x), x, termorder=forward)), n=0..10); # Peter Luschny, Dec 18 2014
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MATHEMATICA
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Table[CoefficientList[2^n*LegendreP[n, x], x], {n, 0, 10}]; Flatten[%]
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PROG
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(PARI) tabl(nn) = for (n=0, nn, print(Vecrev(2^n*pollegendre(n)))); \\ Michel Marcus, Dec 18 2014
(Sage)
if n==0: return [1]
T = [c[0] for c in (2^n*gen_legendre_P(n, 0, x)).coefficients()]
return [0 if is_odd(n+k) else T[k//2] for k in (0..n)]
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CROSSREFS
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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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