User:Juan M. Marquez

math lecturer at Dept of Maths, CUCEI, Universidad de Guadalajara, Mex, with two M.Sc. applied and pures maths. Ph.D. candidate CIMAT A.C.

A two sums of reciprocal problems connected

$2+{\frac {4{\sqrt {3}}\pi }{27}}=1+1+{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{14}}+{\frac {1}{42}}+{\frac {1}{132}}+\cdots$ has the generating function $GF(x)={\frac {2\left({\sqrt {4-x}}(8+x)+12{\sqrt {x}}\arctan {\frac {\sqrt {x}}{\sqrt {4-x}}}\right)}{\sqrt {(4-x)^{5}}}}$

the trick

$\sum _{n=1}^{\infty }{\frac {x^{n}}{2n \choose n}}=1+{\frac {x}{2}}+{\frac {x^{2}}{6}}+{\frac {x^{3}}{20}}+{\frac {x^{4}}{70}}+{\frac {x^{5}}{252}}+\cdots$ multiplied by $\,x$ gives $x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{20}}+{\frac {x^{5}}{70}}+{\frac {x^{6}}{252}}+\cdots$ but differentiating ${\frac {d}{dx}}$ implies

$1+{\frac {2x}{2}}+{\frac {3x}{6}}+{\frac {4x^{3}}{20}}+{\frac {5x^{4}}{70}}+{\frac {6x^{5}}{252}}+\cdots$ , and simplifying throw us $\sum _{n=0}^{\infty }{\frac {(n+1)x^{n}}{2n \choose n}}=1+x+{\frac {x}{2}}+{\frac {x^{3}}{5}}+{\frac {x^{4}}{14}}+{\frac {x^{5}}{42}}+\cdots$ this allows to infer the connection of two G.f's:

${\frac {d}{dx}}\left({\frac {4x\left({\sqrt {4-x}}+{\sqrt {x}}\arcsin({\frac {\sqrt {x}}{2}})\right)}{\sqrt {(4-x)^{3}}}}\right)={\frac {2\left({\sqrt {4-x}}(8+x)+12{\sqrt {x}}\arctan {\frac {\sqrt {x}}{\sqrt {4-x}}}\right)}{\sqrt {(4-x)^{5}}}}$

The function $\scriptstyle {\frac {4\left({\sqrt {4-x}}+{\sqrt {x}}\arcsin({\frac {\sqrt {x}}{2}})\right)}{\sqrt {(4-x)^{3}}}}$ is known since Sprugnoli 2006 in his SUMS OF RECIPROCALS OF THE CENTRAL BINOMIAL COEFFICIENTS and $\scriptstyle {\frac {2\left({\sqrt {4-x}}(8+x)+12{\sqrt {x}}\arctan {\frac {\sqrt {x}}{\sqrt {4-x}}}\right)}{\sqrt {(4-x)^{5}}}}$ was nowhere...

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webpage: Sum of reciprocals of juanmarqz.wordpress.com

NEWS

A164001 counts the number of words ordered by word length on the free product group $\langle a,b\ |\ a^{2}=e,\ b^{3}=e\rangle$

Z2*Z3 =<( a, b :|: a^2, b^3 )>

0 -> e, (1)

1 -> a, b (2)

2 -> ab, ba, bb (3)

3 -> aba, abb, bab, bba (4)

4 -> abab, abba, baba, babb, bbab (5)

5 -> ababa, abbab, babab, babba, bbaba, bbabb (7)

6 -> ababab, abbaba, abbabb, bababa, bababb, babbab, bbabab, bbabba (9)

My TwoCents

at https://oeis.org/A171503 on, as I called: "Siehler Numbers"

at https://oeis.org/A121839 on the sum of reciprocal Catalan Numberrrs

at https://oeis.org/A137243 how many pair of coprimes

at https://oeis.org/A000127 on counting regions