A267844 a(n) = Catalan(n)^2*(4n + 3).

3, 7, 44, 375, 3724, 40572, 470448, 5705271, 71571500, 921922716, 12130541488, 162422308412, 2206718599344, 30354522550000, 422005129502400, 5921371233163575, 83761043464536300, 1193351781764231100, 17110404580326750000, 246734315435589111900

Offset: 0

Author

Ralf Steiner, Jan 21 2016

Keywords

• nonn
• frac

Comments

Numerator of the modified (4n+3) Wallis-Lambert-series-1 with denominator A013709 convergent to 1. Proof: Both the Wallis-Lambert-series-1=4/Pi-1 and the elliptic Euler-series=1-2/Pi are absolutely convergent series. Thus any linear combination of the terms of these series will be also absolutely convergent to the value of the linear combination of these series - in this case to 1. Q.E.D.

Links

• Ralf Steiner, Notiz zur modifizierten Wallis-Lambert-Reihe (in German).

Crossrefs

Cf. A000108, A013709 (denominator).

Programs

Magma

[Catalan(n)^2*(4*n+3):n in [0..20]]; // Vincenzo Librandi, Jan 25 2016

Mathematica

Table[CatalanNumber[n]^2 (4 n + 3), {n, 0, 19}] (* Michael De Vlieger, Jan 24 2016 *)

Formula

a(n) = Catalan(n)^2*(4n + 3).