A157077 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.
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
Offset: 0
Examples
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}.
.
From _Jon E. Schoenfield_, Jul 04 2022: (Start)
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)
Links
- Paul W. Haggard, Some applications of Legendre numbers, International Journal of Mathematics and Mathematical Sciences, vol. 11, Article ID 538097, 8 pages, 1988. See Table 3 p. 412.
- Eric Weisstein's World of Mathematics, Legendre Polynomial
Programs
-
Maple
with(orthopoly):with(PolynomialTools): seq(print(CoefficientList (2^n*P(n, x), x,termorder=forward)),n=0..10); # Peter Luschny, Dec 18 2014
-
Mathematica
Table[CoefficientList[2^n*LegendreP[n, x], x], {n, 0, 10}]; Flatten[%] -
PARI
tabl(nn) = for (n=0, nn, print(Vecrev(2^n*pollegendre(n)))); \\ Michel Marcus, Dec 18 2014
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Sage
def A157077_row(n): 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)] for n in range(9): print(A157077_row(n)) # Peter Luschny, Dec 19 2014
Formula
Row sums are 2^n.
From Peter Luschny, Dec 19 2014: (Start)
T(n,0) = A126869(n).
T(n,n) = A000984(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.
Q(n,1) = A000079(n).
Q(n,2) = A069835(n).
Q(n,3) = A084773(n).
Q(n,4) = A098269(n).
Q(n,5) = A098270(n). (End)
From Fabián Pereyra, Jun 30 2022: (Start)
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)
Extensions
Name clarified and edited by Peter Luschny, Dec 18 2014