A067002 Numerator of Sum_{k=0..n} 2^(k-2*n) * binomial(2*n-2*k,n-k) * binomial(n+k,n).

1, 3, 21, 77, 1155, 4389, 33649, 129789, 4023459, 15646785, 122044923, 477084699, 7474326951, 29322359577, 230389968105, 906200541213, 57090634096419, 225004263791769, 1775033636579511, 7006711723340175, 110706045228774765

Offset: 0

Author

N. J. A. Sloane, Feb 16 2002

Keywords

• nonn
• frac

Comments

Numerator of e(0,n) (see the Maple line).

The generating function of the full fraction is (1-2*x)^(-3/4). - R. J. Mathar, Nov 06 2011

Examples

			1, 3/2, 21/8, 77/16, 1155/128, 4389/256, 33649/1024, 129789/2048, 4023459/32768, ... = A067002/A046161.
		

Links

• V. H. Moll. The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.

Crossrefs

Denominators are in A046161.

Cf. A067001, A223549, A223550.

Programs

Maple

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;

Mathematica

Numerator[Table[Sum[2^(k-2n) Binomial[2n-2k,n-k]Binomial[n+k,n],{k,0,n}],{n,0,30}]] (* Harvey P. Dale, Oct 19 2012 *)

Formula

Numerator of 2^n*Gamma(n + 3/4)/(Gamma(3/4)*n!). - R. J. Mathar, Nov 06 2011

Numerator of integral_{x>0} 1/(x^4 + 1)^(n+1) / (Pi*sqrt(2)). - Jean-François Alcover, Apr 29 2013

From Petros Hadjicostas, May 23 2020: (Start)

If fr(n) = A067002(n)/A046161(n), then fr(n) = P_n(0), where P_n(x) is the Boros-Moll polynomial mentioned in A223549 and A223550 (and whose coefficients are the numbers e(l,n) = A067001(n,n-l)/2^(2*n) that are mentioned in the Maple line below with l = 0..n).

Recurrence for fr(n): 4*n*(n - 1)*fr(n) = 8*(2*n - 1)*(n - 1)*fr(n-1) - (16*(n-1)^2 - 1)*fr(n-2) for n >= 2 with fr(0) = 1 and fr(1) = 3/2. (End)