A001800 -id:A001800 - cp's OEIS frontend

A001790 Numerators in expansion of 1/sqrt(1-x).

1, 1, 3, 5, 35, 63, 231, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 300540195, 583401555, 2268783825, 4418157975, 34461632205, 67282234305, 263012370465, 514589420475, 8061900920775, 15801325804719

Offset: 0

N. J. A. Sloane

Also numerator of e(n-1,n-1) (see Maple line).
Leading coefficient of normalized Legendre polynomial.
Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x).
Also the numerator of binomial(2*n,n)/2^n. - T. D. Noe, Nov 29 2005
This sequence gives the numerators of the Maclaurin series of the Lorentz factor (see Wikipedia link) of 1/sqrt(1-b^2) = dt/dtau where b=u/c is the velocity in terms of the speed of light c, u is the velocity as observed in the reference frame where time t is measured and tau is the proper time. - Stephen Crowley, Apr 03 2007
Truncations of rational expressions like those given by the numerator operator are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n)$ / sigma(2*n) = A056040(2*n) / A060632(2*n+1). Simply said: this sequence is the odd part of the swinging factorial at even indices. - Peter Luschny, Aug 01 2009
It appears that a(n) = A060818(n)*A001147(n)/A000142(n). - James R. Buddenhagen, Jan 20 2010
The convolution of sequence binomial(2*n,n)/4^n with itself is the constant sequence with all terms = 1.
a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). - John M. Campbell, Jul 04 2011
a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2-2*x+2)^n dx. - Leonid Bedratyuk, Nov 17 2012
a(n) = numerator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - Jean-François Alcover, Mar 21 2013
Constant terms for normalized Legendre polynomials. - Tom Copeland, Feb 04 2016
From Ralf Steiner, Apr 07 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums:
a(n)/A060818(n) = (-2)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} a(k)/A060818(k) = -i.
Sum_{k>=0} (-1)^k*a(k)/A060818(k) = 1/sqrt(3).
Sum_{k>=0} (-1)^(k+1)*a(k)/A060818(k) = -1/sqrt(3).
a(n)/A046161(n) = (-1)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} (-1)^k*a(k)/A046161(k) = 1/sqrt(2).
Sum_{k>=0} (-1)^(k+1)*a(k)/A046161(k) = -1/sqrt(2). (End)
a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2+1)^n dx. (n=1 is the Cauchy distribution.) - Harry Garst, May 26 2017
Let R(n, d) = (Product_{j prime to d} Pochhammer(j / d, n)) / n!. Then the numerators of R(n, 2) give this sequence and the denominators are A046161. For d = 3 see A273194/A344402. - Peter Luschny, May 20 2021
Using WolframAlpha, it appears a(n) gives the numerator in the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - Nicholas Juricic, Mar 31 2022
a(n) is the numerator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding denominator is A046161. - Mohammed Yaseen, Jul 29 2023
a(n) is the numerator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding denominator is A101926(n). - Mohammed Yaseen, Sep 19 2023

			1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ...
binomial(2*n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...

P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, 1968; Chap. III, Eq. 4.1.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:6 at page 51.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 102.

Robert G. Wilson v, Table of n, a(n) for n = 0..1666 (first 201 terms from T. D. Noe)
Horst Alzer and Bent Fuglede, Normalized binomial mid-coefficients and power means, Journal of Number Theory, Volume 115, Issue 2, December 2005, Pages 284-294.
C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304062, 1993 (see V_n with N=1).
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table [Annotated scanned copy].
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
S. Łukaszyk and W. Bieniawski, Assembly Theory of Binary Messages, Mathematics, 12(10) (2024), 1600.
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages).
Eric Weisstein's World of Mathematics, Binomial Series.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Wikipedia, Lorentz Factor.

Cf. A000142, A000984, A001147, A001316, A001800, A001801, A001803.
Cf. A005187, A007814, A008316, A046161, A056040, A086117, A094638.
Cf. A101926, A123854, A273194, A344402.
Cf. A060818 (denominator of binomial(2*n,n)/2^n), A061549 (denominators).
Cf. A123854 (denominators).
Cf. A161198 (triangle of coefficients for (1-x)^((-1-2*n)/2)).
Cf. A163590 (odd part of the swinging factorial).
Cf. A001405.
First column and diagonal 1 of triangle A100258.
Bisection of A036069.
Bisections give A061548 and A063079.
Inverse Moebius transform of A180403/A046161.
Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), this sequence (p=-1), A001803 (p=-3), A161199 (p=-5), A161201 (p=-7).

A001790:= func< n | Numerator((n+1)*Catalan(n)/4^n) >;
[A001790(n): n in [0..40]]; // G. C. Greubel, Sep 23 2024

e := proc(l,m) local k; add(2^(k-2*m)*binomial(2*m-2*k,m-k)*binomial(m+k,m)*binomial(k,l),k=l..m); end;
# From Peter Luschny, Aug 01 2009: (Start)
swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i,i=convert(iquo(n,2),base,2))):
a := n -> swing(2*n)/sigma(2*n); # (End)
A001790 := proc(n) binomial(2*n, n)/4^n ; numer(%) ; end proc : # R. J. Mathar, Jan 18 2013

Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]
Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}]   (* John M. Campbell, Jul 04 2011 *)
Numerator[Table[(-2)^n*Sqrt[Pi]/(Gamma[1/2 - n]*Gamma[1 + n]),{n,0,20}]] (* Ralf Steiner, Apr 07 2017 *)
Numerator[Table[Binomial[2n,n]/2^n, {n, 0, 25}]] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Numerator@LegendreP[2 n, 0]*(-1)^n, {n, 0, 25}] (* Andres Cicuttin, Jan 22 2018 *)
A = {1}; Do[A = Append[A, 2^IntegerExponent[n, 2]*(2*n - 1)*A[[n]]/n], {n, 1, 25}]; Print[A] (* John Lawrence, Jul 17 2020 *)

{a(n) = if( n<0, 0, polcoeff( pollegendre(n), n) * 2^valuation((n\2*2)!, 2))};

a(n)=binomial(2*n,n)>>hammingweight(n); \\ Gleb Koloskov, Sep 26 2021

# uses[A000120]
@CachedFunction
def swing(n):
    if n == 0: return 1
    return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
A001790 = lambda n: swing(2*n)/2^A000120(2*n)
[A001790(n) for n in (0..25)]  # Peter Luschny, Nov 19 2012

a(n) = numerator( binomial(2*n,n)/4^n ) (cf. A046161).
a(n) = A000984(n)/A001316(n) where A001316(n) is the highest power of 2 dividing C(2*n, n) = A000984(n). - Benoit Cloitre, Jan 27 2002
a(n) = denominator of (2^n/binomial(2*n,n)). - Artur Jasinski, Nov 26 2011
a(n) = numerator(L(n)), with rational L(n):=binomial(2*n,n)/2^n. L(n) is the leading coefficient of the Legendre polynomial P_n(x).
L(n) = (2*n-1)!!/n! with the double factorials (2*n-1)!! = A001147(n), n >= 0.
Numerator in (1-2t)^(-1/2) = 1 + t + (3/2)t^2 + (5/2)t^3 + (35/8)t^4 + (63/8)t^5 + (231/16)t^6 + (429/16)t^7 + ... = 1 + t + 3*t^2/2! + 15*t^3/3! + 105*t^4/4! + 945*t^5/5! + ... = e.g.f. for double factorials A001147 (cf. A094638). - Tom Copeland, Dec 04 2013
From Ralf Steiner, Apr 08 2017: (Start)
a(n)/A061549(n) = (-1/4)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} a(k)/A061549(k) = 2/sqrt(3).
Sum_{k>=0} (-1)^k*a(k)/A061549(k) = 2/sqrt(5).
Sum_{k>=0} (-1)^(k+1)*a(k)/A061549(k) = -2/sqrt(5).
a(n)/A123854(n) = (-1/2)^n*sqrt(Pi)/(gamma(1/2 - n)*gamma(1 + n)).
Sum_{k>=0} a(k)/A123854(k) = sqrt(2).
Sum_{k>=0} (-1)^k*a(k)/A123854(k) = sqrt(2/3).
Sum_{k>=0} (-1)^(k+1)*a(k)/A123854(k) = -sqrt(2/3). (End)
a(n) = 2^A007814(n)*(2*n-1)*a(n-1)/n. - John Lawrence, Jul 17 2020
Sum_{k>=0} A086117(k+3)/a(k+2) = Pi. - Antonio Graciá Llorente, Aug 31 2024
a(n) = A001803(n)/(2*n+1). - G. C. Greubel, Sep 23 2024

A001803 Numerators in expansion of (1 - x)^(-3/2).

Original entry on oeis.org

1, 3, 15, 35, 315, 693, 3003, 6435, 109395, 230945, 969969, 2028117, 16900975, 35102025, 145422675, 300540195, 9917826435, 20419054425, 83945001525, 172308161025, 1412926920405, 2893136075115, 11835556670925

Offset: 0

N. J. A. Sloane

a(n) is the denominator of the integral from 0 to Pi of (sin(x))^(2*n+1). - James R. Buddenhagen, Aug 17 2008
a(n) is the denominator of (2n)!!/(2*n + 1)!! = 2^(2*n)*n!*n!/(2*n + 1)! (see Andersson). - N. J. A. Sloane, Jun 27 2011
a(n) = (2*n + 1)*A001790(n). A046161(n)/a(n) = 1, 2/3, 8/15, 16/35, 128/315, 256/693, ... is binomial transform of Madhava-Gregory-Leibniz series for Pi/4 (i.e., 1 - 1/3 + 1/5 - 1/7 + ... ). See A173384 and A173396. - Paul Curtz, Feb 21 2010
a(n) is the denominator of Integral_{x=-oo..oo} sech(x)^(2*n+2) dx. The corresponding numerator is A101926(n). - Mohammed Yaseen, Jul 25 2023

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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:9 at page 51.

T. D. Noe, Table of n, a(n) for n = 0..200
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Mats Erik Andersson, Das Flaviussche Sieb, Acta Arith., 85 (1998), 301-307.
Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020.
Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Eric Weisstein's World of Mathematics, Heads-Minus-Tails Distribution, Random Walk--1-Dimensional, Circle Line Picking.

The denominator is given in A046161.
Largest odd divisors of A001800, A002011, A002457, A005430, A033876, A086228.
Bisection of A004731, A004735, A086116.
Second column of triangle A100258.
Cf. A161199, A161201, A173384, A173396.
Cf. A002596 (numerators in expansion of (1-x)^(1/2)).
Cf. A161198 (triangle related to the series expansions of (1-x)^((-1-2*n)/2)).
A163590 is the odd part of the swinging factorial, A001790 at even indices. - Peter Luschny, Aug 01 2009

A001803(n) = sum(<<(A001790(k), A005187(n) - A005187(k)) for k in 0:n) # Peter Luschny, Oct 03 2019

A001803:= func< n | Numerator(Binomial(n+2,2)*Catalan(n+1)/4^n) >;
[A001803(n): n in [0..30]]; // G. C. Greubel, Apr 27 2025

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i,i= convert(iquo(n,2),base,2))):
a := n -> swing(2*n+1)/sigma(2*n+1); # Peter Luschny, Aug 01 2009
A001803 := proc(n) (2*n+1)*binomial(2*n,n)/4^n ; numer(%) ; end proc: # R. J. Mathar, Jul 06 2011
a := n -> denom(Pi*binomial(n, -1/2)): seq(a(n), n = 0..22); # Peter Luschny, Dec 06 2024

Numerator/@CoefficientList[Series[(1-x)^(-3/2),{x,0,25}],x]  (* Harvey P. Dale, Feb 19 2011 *)
Table[Denominator[Beta[1, n + 1, 1/2]], {n, 0, 22}] (* Gerry Martens, Nov 13 2016 *)

a(n) = numerator((2*n+1)*binomial(2*n,n)/(4^n)); \\ Altug Alkan, Sep 06 2018

def A001803(n): return numerator((n+1)*binomial(2*n+2,n+1)/2^(2*n+1))
print([A001803(n) for n in range(31)]) # G. C. Greubel, Apr 27 2025

a(n) = (2*n + 1)! /(n!^2*2^A000120(n)) = (n + 1)*binomial(2*n+2,n+1)/2^(A000120(n)+1). - Ralf Stephan, Mar 10 2004
From Johannes W. Meijer, Jun 08 2009: (Start)
a(n) = numerator( (2*n+1)*binomial(2*n,n)/(4^n) ).
(1 - x)^(-3/2) = Sum_{n>=0} ((2*n+1)*binomial(2*n,n)/4^n)*x^n. (End)
Truncations of rational expressions like those given by the numerator or denominator operators are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n+1)$ / sigma(2*n+1) = A056040(2*n+1) / A060632(2*n+2). Simply said: This sequence gives the odd part of the swinging factorial at odd indices. - Peter Luschny, Aug 01 2009
a(n) = denominator(Pi*binomial(n, -1/2)). - Peter Luschny, Dec 06 2024

A008316 Triangle of coefficients of Legendre polynomials P_n (x).

Original entry on oeis.org

1, 1, -1, 3, -3, 5, 3, -30, 35, 15, -70, 63, -5, 105, -315, 231, -35, 315, -693, 429, 35, -1260, 6930, -12012, 6435, 315, -4620, 18018, -25740, 12155, -63, 3465, -30030, 90090, -109395, 46189, -693, 15015, -90090, 218790, -230945, 88179, 231, -18018, 225225, -1021020, 2078505, -1939938, 676039

Offset: 0

N. J. A. Sloane

			Triangle starts:
   1;
   1;
  -1,   3;
  -3,   5;
   3, -30, 35;
  15, -70, 63;
  ...
P_5(x) = (15*x - 70*x^3 + 63*x^5)/8 so T(5, ) = (15, -70, 63). P_6(x) = (-5 + 105*x^2 - 315*x^4 + 231*x^6)/16 so T(6, ) = (-5, 105, -315, 231). - _Michael Somos_, Oct 24 2002

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.

T. D. Noe, Rows n=0..100 of triangle, flattened
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].
T. Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera, see the Additional Notes section, 2015.
H. N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
Eric Weisstein's World of Mathematics, Legendre Polynomial

Cf. A001790, A001800, A001801.
With zeros: A100258.
Cf. A121448.

Flatten[Table[(LegendreP[i, x]/.{Plus->List, x->1})Max[ Denominator[LegendreP[i, x]/.{Plus->List, x->1}]], {i, 0, 12}]]

{T(n, k) = if( n<0, 0, polcoeff( pollegendre(n) * 2^valuation( (n\2*2)!, 2), n%2 + 2*k))}; /* Michael Somos, Oct 24 2002 */

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002

A100258 Triangle of coefficients of normalized Legendre polynomials, with increasing exponents.

Original entry on oeis.org

1, 0, 1, -1, 0, 3, 0, -3, 0, 5, 3, 0, -30, 0, 35, 0, 15, 0, -70, 0, 63, -5, 0, 105, 0, -315, 0, 231, 0, -35, 0, 315, 0, -693, 0, 429, 35, 0, -1260, 0, 6930, 0, -12012, 0, 6435, 0, 315, 0, -4620, 0, 18018, 0, -25740, 0, 12155, -63, 0, 3465, 0, -30030, 0, 90090, 0, -109395, 0, 46189

Offset: 0

Ralf Stephan, Nov 13 2004

For a relation to Jacobi quartic elliptic curves, see the MathOverflow link. For a self-convolution of the polynomials relating them to the Chebyshev and Fibonacci polynomials, see A049310 and A053117. For congruences and connections to other polynomials (Jacobi, Gegenbauer, and Chebyshev) see the Allouche et al. link. For relations to elliptic cohomology and modular forms, see references in Copeland link.- Tom Copeland, Feb 04 2016

			Triangle begins:
   1;
   0,   1;
  -1,   0,     3;
   0,  -3,     0,   5;
   3,   0,   -30,   0,   35;
   0,  15,     0, -70,    0,   63;
  -5,   0,   105,   0, -315,    0,    231;
   0, -35,     0, 315,    0, -693,      0, 429;
  35,   0, -1260,   0, 6930,    0, -12012,   0, 6435;
  ...

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.

T. D. Noe, Rows n=0..100 of triangle, flattened
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].
J. Allouche and G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, p. 21-49.
Tom Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera
H. N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
MathOverflow, Geometric picture of invariant differential of an elliptic curve, Dec 4 2011.

Without zeros: A008316. Row sums are A060818.
Columns (with interleaved zeros and signs) include A001790, A001803, A100259. Diagonals include A001790, A001800, A001801, A001802.
Cf. A049310, A005187, A049600, A053117.

row[n_] := CoefficientList[ LegendreP[n, x], x]*2^IntegerExponent[n!, 2]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 15 2015 *)

a(k,n)=polcoeff(pollegendre(k,x),n)*2^valuation(k!,2)

from mpmath import *
mp.dps=20
def a007814(n):
    return 1 + bin(n - 1)[2:].count('1') - bin(n)[2:].count('1')
for n in range(11):
    y=2**sum(a007814(i) for i in range(2, n+1))
    l=chop(taylor(lambda x: legendre(n, x), 0, n))
    print([int(i*y) for i in l]) # Indranil Ghosh, Jul 02 2017

The n-th normalized Legendre polynomial is generated by 2^(-n-a(n)) (d/dx)^n (x^2-1)^n / n! with a(n) = A005187(n/2) for n even and a(n) = A005187((n-1)/2) for n odd. The non-normalized polynomials have the o.g.f. 1 / sqrt(1 - 2xz + z^2). - Tom Copeland, Feb 07 2016

The consecutive nonzero entries in the m-th row are, in order, (c+b)!/(c!(m-b)!(2b-m)!*A048896(m-1)) with sign (-1)^b where c = m/2-1, m/2, m/2+1, ..., (m-1) and b = c+1 if m is even and sign (-1)^c with c = (m-1)/2, (m-1)/2+1, (m-1)/2+2, ..., (m-1) with b = c+1 if m is odd. For the 9th row the 5 consecutive nonzero entries are 315, -4620, 18018, -25740, 12155 given by c = 4,5,6,7,8 and b = 5,6,7,8,9. - Richard Turk, Aug 22 2017

A001796 Coefficients of Legendre polynomials.

Original entry on oeis.org

1, 3, 27, 143, 3315, 20349, 260015, 1710855, 92116035, 631165425, 8775943605, 61750730457, 1755702867191, 12587970424449, 181858466731095, 1322239639929719, 154702037871777123, 1137023085979691001, 16789716964765636633

Offset: 0

N. J. A. Sloane

nonn

Numerators in expansion of c(x)^(3/2), c(x) the g.f. of A000108. - Gerald McGarvey, Oct 07 2008

Coefficient of Legendre_1(x) when x^n is written in term of Legendre polynomials. - Michel Marcus, May 28 2013

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..830
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.

Cf. A001795, A001797, A001798, A001799, A001800, A001801, A001802.

Cf. A000108.

A001796:= func< n | Numerator(3*(n+1)*Catalan(2*n+1)/(4^n*(2*n+3))) >;
[A001796(n): n in [0..25]]; // G. C. Greubel, Apr 23 2025

Table[Numerator[3*Binomial[2*n+1/2, n]/(2*n+3)], {n,0,30}] (* G. C. Greubel, Apr 23 2025 *)

my(x='x+O('x^30)); apply(numerator, Vec(((1-sqrt(1-4*x))/(2*x))^(3/2))) \\ Michel Marcus, Feb 04 2022

a(n)=numerator(3*binomial(2*n+1/2, n)/(2*n+3)) \\ Tani Akinari, Oct 31 2024

def A001796(n): return numerator(3*binomial(2*n+1/2, n)/(2*n+3))
print([A001796(n) for n in range(31)]) # G. C. Greubel, Apr 23 2025

Numerators of g.f. ((1-sqrt(1-4*x))/(2*x))^(3/2). - Sean A. Irvine, Nov 27 2012

a(n) = numerator(3*binomial(2*n+1/2, n)/(2*n+3)). - Tani Akinari, Oct 31 2024

More terms from Sean A. Irvine, Nov 27 2012

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A001790 Numerators in expansion of 1/sqrt(1-x).

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A001803 Numerators in expansion of (1 - x)^(-3/2).

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A008316 Triangle of coefficients of Legendre polynomials P_n (x).

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A100258 Triangle of coefficients of normalized Legendre polynomials, with increasing exponents.

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A001796 Coefficients of Legendre polynomials.

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