Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/BinomialSeries.html">Binomial Series</a>.

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

reviewed

approved

proposed

reviewed

editing

proposed

W. G. Bickley and J. C. P. Miller, <a href="/A002551/a002551.pdf">Numerical differentiation near the limits of a difference table</a> [Annotated scanned copy].

J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages).

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSeries.html">Binomial Series</a>.

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

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

approved

editing

reviewed

approved

proposed

reviewed

editing

proposed

Also numerator of binomial(2n,n)/4^n (cf. A046161).

Also the numerator of binomial(2n,2*n,n)/2^n. - T. D. Noe, Nov 29 2005

The convolution of sequence binomial(2n,2*n,n)/4^n with itself is the constant sequence with all terms = 1.

a(n) = denominator of 2^n*n!*n!/(2*n)!. - Artur Jasinski, Nov 26 2011

a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2-2x2*x+2)^n dx. - Leonid Bedratyuk, Nov 17 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(2n, 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) = A001803(n)/(2*n+1). - G. C. Greubel, Sep 23 2024

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

(Magma)

A001790:= func< n | Numerator((n+1)*Catalan(n)/4^n) >;

[A001790(n): n in [0..40]]; // G. C. Greubel, Sep 23 2024

Cf. A000142, A000984, A001147, A001316, A001800, A001801, A008316, A046161A001803.

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. A005187, A060818(n)= denominator(L(n)). Bisections give A061548 and A063079.

From Johannes W. Meijer, Jun 08 2009: (Start)

Cf. A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].

A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n.

(End)

A163590 is the odd part of the swinging factorial, A001803 at odd indices.

Inverse Moebius transform of A180403/A046161. - _Mats Granvik_, Sep 04, 2010

Cf. A123854 (denominators), A061549 (denominators). - Ralf Steiner, Apr 08 2017

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).

approved

editing