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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..65.

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

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)

EXAMPLE

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}.

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[%]

PROG

(PARI) tabl(nn) = for (n=0, nn, print(Vecrev(2^n*pollegendre(n)))); \\ Michel Marcus, Dec 18 2014

(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

CROSSREFS

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

Sequence in context: A221408 A221396 A221337 * A185896 A076256 A127467

Adjacent sequences:  A157074 A157075 A157076 * A157078 A157079 A157080

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Feb 22 2009

EXTENSIONS

Name clarified and edited by Peter Luschny, Dec 18 2014

STATUS

approved

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Last modified April 17 05:12 EDT 2021. Contains 343059 sequences. (Running on oeis4.)