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%I #53 Jan 05 2023 03:20:39
%S 1,1,1,4,5,1,7,8,3,5,11,4,13,7,5,16,17,3,19,20,7,11,23,8,25,13,9,28,
%T 29,5,31,32,11,17,35,12,37,19,13,40,41,7,43,44,15,23,47,16,49,25,17,
%U 52,53,9,55,56,19,29,59,20,61,31,21,64,65,11,67,68,23,35,71,24
%N a(n) is the numerator of 2*n / ((n+2) * (n+3)).
%C Previous name was: Numerators of the third row of the Akiyama-Tanigawa algorithm (or transformation) applied to A001008(n+1)/A002805(n+1).
%C Successive rows:
%C 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, ...;
%C -1/3, -1/2, -3/5, -2/3, -5/7, -3/4, -7/9, -4/5, ... = A026741(n+1)/A026741(n+3);
%C 1/6, 1/5, 1/5, 4/21, 5/28, 1/6, 7/45, 8/55, 3/22, ...;
%C -1/30, 0, ...;
%C -1/30.
%C First column denominators: 2,3,6,30,30,... = A051717(n+1).
%C A001008(n)/A002805(n) is the inverse Akiyama-Tanigawa transformation applied to A027641(n)/A027642(n). A051716(n)/A051717(n) comes from 0 followed by A164555(n)/A027642(n). Then, from the two Bernoulli numbers.
%H G. C. Greubel, <a href="/A237109/b237109.txt">Table of n, a(n) for n = 1..2500</a>
%F a(n) = -a(-n) for all n in Z. - _Michael Somos_, Aug 01 2017
%F From _Amiram Eldar_, Nov 17 2022: (Start)
%F Multiplicative with a(2) = 1, a(2^e) = 2^e for e > 1, a(3^e) = 3^(e-1), and a(p^e) = p^e for p >= 5.
%F Sum_{k=1..n} a(k) ~ (49/144) * n^2. (End)
%F Dirichlet g.f.: zeta(s-1)*(1-1/2^s+2/4^s)*(1-2/3^s). - _Amiram Eldar_, Jan 05 2023
%t a[1, n_] := HarmonicNumber[n+1]; a[n_, m_] := a[n, m] = m*(a[n-1, m]-a[n-1, m+1]); Table[a[3, m] // Numerator, {m, 1, 72}] (* _Jean-François Alcover_, Feb 11 2014 *)
%t a[ n_] := n / {1, 2, 3, 1, 1, 6, 1, 1, 3, 2, 1, 3}[[Mod[n, 12, 1]]]; (* _Michael Somos_, Aug 01 2017 *)
%o (PARI) {a(n) = if( n<0, -a(-n), numerator( 2*n / ((n+2) * (n+3))))}; /* _Michael Somos_, Aug 01 2017 */
%o (Magma) [Numerator(2*n/((n+2)*(n+3))): n in [1..50]]; // _G. C. Greubel_, Aug 07 2018
%K nonn,frac,mult
%O 1,4
%A _Paul Curtz_, Feb 03 2014
%E New name using Somos's Pari code from _Joerg Arndt_, May 27 2018
%E Keyword:mult added by _Andrew Howroyd_, Jul 31 2018
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