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A267844 a(n) = Catalan(n)^2*(4n + 3). 1
3, 7, 44, 375, 3724, 40572, 470448, 5705271, 71571500, 921922716, 12130541488, 162422308412, 2206718599344, 30354522550000, 422005129502400, 5921371233163575, 83761043464536300, 1193351781764231100, 17110404580326750000, 246734315435589111900 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

REFERENCES

Filed on Documenta Mathematica (17.01.2016).

LINKS

Table of n, a(n) for n=0..19.

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

FORMULA

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

EXAMPLE

For n=3 the a(3)=375.

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000108, A013709(denominator).

Sequence in context: A253576 A019011 A036842 * A041349 A041016 A301324

Adjacent sequences:  A267841 A267842 A267843 * A267845 A267846 A267847

KEYWORD

nonn,frac

AUTHOR

Ralf Steiner, Jan 21 2016

STATUS

approved

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Last modified December 17 02:28 EST 2018. Contains 318192 sequences. (Running on oeis4.)