A100258 -id:A100258 - cp's OEIS frontend

A372271 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.

2, 3, 8, 6, 1, 9, 1, 8, 6, 0, 8, 3, 1, 9, 6, 9, 0, 8, 6, 3, 0, 5, 0, 1, 7, 2, 1, 6, 8, 0, 7, 1, 1, 9, 3, 5, 4, 1, 8, 6, 1, 0, 6, 3, 0, 1, 4, 0, 0, 2, 1, 3, 5, 0, 1, 8, 1, 3, 9, 5, 1, 6, 4, 5, 7, 4, 2, 7, 4, 9, 3, 4, 2, 7, 5, 6, 3, 9, 8, 4, 2, 2, 4, 9, 2, 2, 4

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.238619186083196908630501721680711935418610630140021350181395...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[6, #] &, 4], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Smallest positive root of 231*x^6 - 315*x^4 + 105*x^2 - 5 = 0.

A372272 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.

Original entry on oeis.org

6, 6, 1, 2, 0, 9, 3, 8, 6, 4, 6, 6, 2, 6, 4, 5, 1, 3, 6, 6, 1, 3, 9, 9, 5, 9, 5, 0, 1, 9, 9, 0, 5, 3, 4, 7, 0, 0, 6, 4, 4, 8, 5, 6, 4, 3, 9, 5, 1, 7, 0, 0, 7, 0, 8, 1, 4, 5, 2, 6, 7, 0, 5, 8, 5, 2, 1, 8, 3, 4, 9, 6, 6, 0, 7, 1, 4, 3, 1, 0, 0, 9, 4, 4, 2, 8, 6

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.661209386466264513661399595019905347006448564395170070814526...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[6, #] &, 5], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Middle positive root of 231*x^6 - 315*x^4 + 105*x^2 - 5 = 0.

A372273 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.

Original entry on oeis.org

9, 3, 2, 4, 6, 9, 5, 1, 4, 2, 0, 3, 1, 5, 2, 0, 2, 7, 8, 1, 2, 3, 0, 1, 5, 5, 4, 4, 9, 3, 9, 9, 4, 6, 0, 9, 1, 3, 4, 7, 6, 5, 7, 3, 7, 7, 1, 2, 2, 8, 9, 8, 2, 4, 8, 7, 2, 5, 4, 9, 6, 1, 6, 5, 2, 6, 6, 1, 3, 5, 0, 0, 8, 4, 4, 2, 0, 0, 1, 9, 6, 2, 7, 6, 2, 8, 8

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.932469514203152027812301554493994609134765737712289824872549...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[6, #] &, 6], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Largest positive root of 231*x^6 - 315*x^4 + 105*x^2 - 5 = 0.

A372274 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

4, 0, 5, 8, 4, 5, 1, 5, 1, 3, 7, 7, 3, 9, 7, 1, 6, 6, 9, 0, 6, 6, 0, 6, 4, 1, 2, 0, 7, 6, 9, 6, 1, 4, 6, 3, 3, 4, 7, 3, 8, 2, 0, 1, 4, 0, 9, 9, 3, 7, 0, 1, 2, 6, 3, 8, 7, 0, 4, 3, 2, 5, 1, 7, 9, 4, 6, 6, 3, 8, 1, 3, 2, 2, 6, 1, 2, 5, 6, 5, 5, 3, 2, 8, 3, 1, 2

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.405845151377397166906606412076961463347382014099370126387043...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | this sequence, A372275, A372276 | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 5], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.1, 0.5, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ A.H.M. Smeets, May 31 2025

Smallest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

A372275 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

7, 4, 1, 5, 3, 1, 1, 8, 5, 5, 9, 9, 3, 9, 4, 4, 3, 9, 8, 6, 3, 8, 6, 4, 7, 7, 3, 2, 8, 0, 7, 8, 8, 4, 0, 7, 0, 7, 4, 1, 4, 7, 6, 4, 7, 1, 4, 1, 3, 9, 0, 2, 6, 0, 1, 1, 9, 9, 5, 5, 3, 5, 1, 9, 6, 7, 4, 2, 9, 8, 7, 4, 6, 7, 2, 1, 8, 0, 5, 1, 3, 7, 9, 2, 8, 2, 6

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.741531185599394439863864773280788407074147647141390260119955...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | A372274, this sequence, A372276 | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 6], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.6, 0.8, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ A.H.M. Smeets, May 31 2025

Middle positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

A372276 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

9, 4, 9, 1, 0, 7, 9, 1, 2, 3, 4, 2, 7, 5, 8, 5, 2, 4, 5, 2, 6, 1, 8, 9, 6, 8, 4, 0, 4, 7, 8, 5, 1, 2, 6, 2, 4, 0, 0, 7, 7, 0, 9, 3, 7, 6, 7, 0, 6, 1, 7, 7, 8, 3, 5, 4, 8, 7, 6, 9, 1, 0, 3, 9, 1, 3, 0, 6, 3, 3, 3, 0, 3, 5, 4, 8, 4, 0, 1, 4, 0, 8, 0, 5, 7, 3, 0

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.949107912342758524526189684047851262400770937670617783548769...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | A372274, A372275, this sequence | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 7], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.8, 1.0, 429*x^6 - 693*x^4 + 315*x^ - 35) \\ A.H.M. Smeets, May 31 2025

Largest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

A001801 Coefficients of Legendre polynomials.

Original entry on oeis.org

3, 15, 105, 315, 6930, 18018, 90090, 218790, 2078505, 4849845, 22309287, 50702925, 1825305300, 4071834900, 18032411700, 39671305740, 347123925225, 755505013725, 3273855059475, 7064634602025, 121511715154830, 260382246760350, 1112542327066950, 2370198870707850, 20146690401016725

Offset: 0

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.

Cf. A001790, A001796, A008316, A011371.
Bisection of A004733.
Diagonal 3 of triangle A100258.

A001801:= func< n | 3*Binomial(n+3,3)*Catalan(n+2)*2^(Valuation(Factorial(n+4),2)-n-4) >;
[A001801(n): n in [0..30]]; // G. C. Greubel, Apr 26 2025

A001801[n_]:= 3*2^(2*n+1)*Binomial[n+3/2, n]/2^DigitCount[n+4,2,1];
Table[A001801[n], {n,0,40}] (* G. C. Greubel, Apr 26 2025 *)

a(n)=if(n<0,0,polcoeff(pollegendre(n+4),n)*2^valuation((n\2*2+4)!,2))

def A001801(n): return 3*2^(n-3)*binomial(n+3/2,n)*2^valuation(factorial(n+4), 2)
print([A001801(n) for n in range(31)]) # G. C. Greubel, Apr 26 2025

a(n) = 3*2^(n-3)*binomial(n + 3/2, n)*2^A011371(n+4). - G. C. Greubel, Apr 26 2025

More terms from Michael Somos, Oct 25 2002

A001800 Coefficients of Legendre polynomials.

Original entry on oeis.org

1, 3, 30, 70, 315, 693, 12012, 25740, 109395, 230945, 1939938, 4056234, 16900975, 35102025, 1163381400, 2404321560, 9917826435, 20419054425, 167890003050, 344616322050, 1412926920405, 2893136075115, 47342226683700, 96742811049300, 395033145117975

Offset: 0

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Legendre Polynomial, eq. 28.

Cf. A001790, A001796, A001803, A008316.

Diagonal 2 of triangle A100258.

A001800:= func< n | (n+1)*(n+2)*Catalan(n+1)/2^(&+Intseq(n+2, 2)) >;
[A001800(n): n in [0..30]]; // G. C. Greubel, Apr 25 2025

wt:= proc(n) local m, r; m:=n; r:=0;
       while m>0 do r:= r+irem(m, 2, 'm') od; r
     end:
a:= n-> (n+1) *binomial(2*n+2, n+1)/2^wt(n+2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 29 2013

a[n_] := (n+1)*Binomial[2*n+2, n+1]/2^DigitCount[n+2, 2, 1]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 13 2014 *)

a(n)=if(n<0,0,-polcoeff(pollegendre(n+2),n)*2^valuation((n\2*2)!,2))

def A001800(n): return (n+1)*binomial(2*n+2,n+1)//2^sum((n+2).digits(2))
print([A001800(n) for n in range(31)]) # G. C. Greubel, Apr 25 2025

a(n) = (n+1) * C(2n+2, n+1) / 2^A000120(n+2).

More terms from Michael Somos, Oct 25 2002

A178301 Triangle T(n,k) = binomial(n,k)*binomial(n+k+1,n+1) read by rows, 0 <= k <= n.

Original entry on oeis.org

1, 1, 3, 1, 8, 10, 1, 15, 45, 35, 1, 24, 126, 224, 126, 1, 35, 280, 840, 1050, 462, 1, 48, 540, 2400, 4950, 4752, 1716, 1, 63, 945, 5775, 17325, 27027, 21021, 6435, 1, 80, 1540, 12320, 50050, 112112, 140140, 91520, 24310, 1, 99, 2376, 24024, 126126, 378378, 672672, 700128, 393822, 92378

Offset: 0

Alford Arnold, May 30 2010

Antidiagonal sums are given by A113682. - Johannes W. Meijer, Mar 24 2013
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial binomial(x+n,n)*binomial(x+n,n-1) in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022
Chapoton's observation above is correct: the precise expansion is  binomial(x+n,n)*binomial(x+n,n-1) = Sum_{k = 0..n-1} (-1)^k*T(n-1,n-1-k)*binomial(x+2*n-1-k,2*n-1-k), as can be verified using the WZ algorithm. For example, n = 4 gives binomial(x+4,4)*binomial(x+4,3)  = 35*binomial(x+7,7) - 45*binomial(x+6,6) + 15*binomial(x+5,5) - binomial(x+4,4). - Peter Bala, Jun 24 2023

			n=0: 1;
n=1: 1,  3;
n=2: 1,  8,  10;
n=3: 1, 15,  45,   35;
n=4: 1, 24, 126,  224,   126;
n=5: 1, 35, 280,  840,  1050,   462;
n=6: 1, 48, 540, 2400,  4950,  4752,  1716;
n=7: 1, 63, 945, 5775, 17325, 27027, 21021, 6435;

David A. Corneth, Table of n, a(n) for n = 0..10010
Author?, Norm of a continuous function, dxdy.ru (in Russian).

Cf. A007318, A047781 (row sums), A178300, A182626, A123160, A132813.

A178301 := proc(n,k)
        binomial(n,k)*binomial(n+k+1,n+1) ;
end proc: # R. J. Mathar, Mar 24 2013
R := proc(n) add((-1)^(n+k)*(2*k+1)*orthopoly:-P(k,2*x+1)/(n+1), k=0..n) end:
for n from 0 to 6 do seq(coeff(R(n), x, k), k=0..n) od; # Peter Luschny, Aug 25 2021

Flatten[Table[Binomial[n,k]Binomial[n+k+1,n+1],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Aug 23 2014 *)

create_list(binomial(n,k)*binomial(n+k+1,n+1),n,0,12,k,0,n); /* Emanuele Munarini, Dec 16 2016 */

R(n,x) = sum(k=0,n, (-1)^(n+k) * (2*k+1) * pollegendre(k,2*x+1)) / (n+1); \\ Max Alekseyev, Aug 25 2021

T(n,k) = A007318(n,k) * A178300(n+1,k+1).
From Peter Bala, Jun 18 2015: (Start)
n-th row polynomial R(n,x) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k+1,n+1)*x^k = Sum_{k = 0..n} (-1)^(n+k)*binomial(n+1,k+1)*binomial(n+k+1,n+1)*(1 + x)^k.
Recurrence: (2*n - 1)*(n + 1)*R(n,x) = 2*(4*n^2*x + 2*n^2 - x - 1)*R(n-1,x) - (2*n + 1)(n - 1)*R(n-2,x) with R(0,x) = 1, R(1,x) = 1 + 3*x.
A182626(n) = -R(n-1,-2) for n >= 1. (End)
From Peter Bala, Jul 20 2015: (Start)
n-th row polynomial R(n,x) = Jacobi_P(n,0,1,2*x + 1).
(1 + x)*R(n,x) gives the row polynomials of A123160. (End)
G.f.: (1+x-sqrt(1-2*x+x^2-4*x*y))/(2*(1+y)*x*sqrt(1-2*x+x^2-4*x*y)). - Emanuele Munarini, Dec 16 2016
R(n,x) = Sum_{k=0..n} (-1)^(n+k)*(2*k+1)*P(k,2*x+1)/(n+1), where P(k,x) is the k-th Legendre polynomial (cf. A100258) and P(k,2*x+1) is the k-th shifted Legendre polynomial (cf. A063007). - Max Alekseyev, Jun 28 2018; corrected by Peter Bala, Aug 08 2021
Polynomial g(n,x) = R(n,-x)/(n+1) delivers the maximum of f(1)^2/(Integral_{x=0..1} f(x)^2 dx) over all polynomials f(x) with real coefficients and deg(f(x)) <= n. This maximum equals (n+1)^2. See dxdy.ru link. - Max Alekseyev, Jun 28 2018

A001802 Coefficients of Legendre polynomials.

Original entry on oeis.org

5, 35, 1260, 4620, 30030, 90090, 1021020, 2771340, 14549535, 37182145, 1487285800, 3650610600, 17644617900, 42075627300, 396713057400, 925663800600, 4281195077775, 9821565178425, 178970743251300, 405039050516100, 1822675727322450, 4079321865912150

Offset: 0

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Diagonal 4 of triangle A100258.

Cf. A001796, A011371.

A001802:= func< n | Binomial(n+4,4)*Catalan(n+3)*2^(Valuation(Factorial(n+6),2)-n-4) >;
[A001802(n): n in [0..30]]; // G. C. Greubel, Apr 26 2025

A001802[n_]:= 5*4^(n+1)*Binomial[n+5/2,n]/2^DigitCount[n+6,2,1];
Table[A001802[n], {n,0,30}] (* G. C. Greubel, Apr 26 2025 *)

a(n)= - polcoeff(pollegendre(n+6), n)*2^valuation((n\2*2+6)!, 2) \\ Michel Marcus, May 29 2013

def A001802(n): return 5*2^(n-4)*binomial(n+5/2,n)*2^valuation(factorial(n+6), 2)
print([A001802(n) for n in range(31)]) # G. C. Greubel, Apr 26 2025

a(n) = 5*2^(n-4)*binomial(n+5/2, n)*2^A011371(n+6). - G. C. Greubel, Apr 26 2025

More terms from Michel Marcus, Feb 02 2015

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A372271 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.

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A372272 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.

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A372273 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.

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A372274 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.

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A372275 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.

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A372276 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.

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A001801 Coefficients of Legendre polynomials.

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A001800 Coefficients of Legendre polynomials.

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