A157077 - OEIS

%I #41 Jul 13 2022 14:52:31

%S 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,

%T -1260,0,924,0,-280,0,2520,0,-5544,0,3432,70,0,-2520,0,13860,0,-24024,

%U 0,12870,0,1260,0,-18480,0,72072,0,-102960,0,48620,-252,0,13860,0,-120120,0,360360,0,-437580,0,184756

%N 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.

%H Paul W. Haggard, <a href="https://doi.org/10.1155/S0161171288000481">Some applications of Legendre numbers</a>, International Journal of Mathematics and Mathematical Sciences, vol. 11, Article ID 538097, 8 pages, 1988. See Table 3 p. 412.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>

%F Row sums are 2^n.

%F From _Peter Luschny_, Dec 19 2014: (Start)

%F T(n,0) = A126869(n).

%F T(n,n) = A000984(n).

%F T(n,1) = (-1)^floor(n/2)*A005430(floor(n/2)+1) if n is odd else 0.

%F Let Q(n, x) = 2^n*P(n, x).

%F Q(n,0) = (-1)^floor(n/2)*A126869(floor(n/2)) if n is even else 0.

%F Q(n,1) = A000079(n).

%F Q(n,2) = A069835(n).

%F Q(n,3) = A084773(n).

%F Q(n,4) = A098269(n).

%F Q(n,5) = A098270(n). (End)

%F From _Fabián Pereyra_, Jun 30 2022: (Start)

%F n*T(n,k) = 2*(2*n-1)*T(n-1,k-1) - 4*(n-1)*T(n-2,k).

%F T(n,k) = (-1)^floor((n-k)/2)*binomial(n+k,k)*binomial(n,floor((n-k)/2))*(1+(-1)^(n-k))/2.

%F 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)

%e 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:

%e {1},

%e {0, 2},

%e {-2, 0, 6},

%e {0, -12, 0, 20},

%e {6, 0, -60, 0, 70},

%e {0, 60, 0, -280, 0, 252},

%e {-20, 0, 420, 0, -1260, 0, 924},

%e {0, -280, 0, 2520, 0, -5544, 0, 3432},

%e {70, 0, -2520, 0, 13860, 0, -24024, 0, 12870},

%e {0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620},

%e {-252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756}.

%e .

%e From _Jon E. Schoenfield_, Jul 04 2022: (Start)

%e As a right-aligned triangle:

%e                                                              1;

%e                                                      0,      2;

%e                                                  -2, 0,      6;

%e                                          0,     -12, 0,     20;

%e                                       6, 0,     -60, 0,     70;

%e                               0,     60, 0,    -280, 0,    252;

%e                          -20, 0,    420, 0,   -1260, 0,    924;

%e                   0,    -280, 0,   2520, 0,   -5544, 0,   3432;

%e               70, 0,   -2520, 0,  13860, 0,  -24024, 0,  12870;

%e         0,  1260, 0,  -18480, 0,  72072, 0, -102960, 0,  48620;

%e   -252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756. (End)

%p with(orthopoly):with(PolynomialTools): seq(print(CoefficientList (2^n*P(n, x), x,termorder=forward)),n=0..10); # _Peter Luschny_, Dec 18 2014

%t Table[CoefficientList[2^n*LegendreP[n, x], x], {n, 0, 10}]; Flatten[%]

%o (PARI) tabl(nn) = for (n=0, nn, print(Vecrev(2^n*pollegendre(n)))); \\ _Michel Marcus_, Dec 18 2014

%o (Sage)

%o def A157077_row(n):

%o     if n==0: return [1]

%o     T = [c[0] for c in (2^n*gen_legendre_P(n, 0, x)).coefficients()]

%o     return [0 if is_odd(n+k) else T[k//2] for k in (0..n)]

%o for n in range(9): print(A157077_row(n)) # _Peter Luschny_, Dec 19 2014

%Y Cf. A100258, A126869, A000984, A005430, A000079, A069835, A084773, A098269, A098270.

%K tabl,sign

%O 0,3

%A _Roger L. Bagula_, Feb 22 2009

%E Name clarified and edited by _Peter Luschny_, Dec 18 2014