A344850 a(n) is the denominator of Catalan-Daehee number d(n).
1, 1, 3, 3, 15, 5, 105, 105, 63, 315, 3465, 495, 6435, 9009, 15015, 15015, 255255, 23205, 37791, 188955, 101745, 1119195, 25741485, 572033, 42902475, 79676025, 42181425, 42181425, 155687805, 40970475, 1270084725, 1270084725, 665282475, 173996955, 6089893425, 794333925
Offset: 0
Links
- Dae San Kim and Taekyun Kim, A new approach to Catalan numbers using differential equations, Russ. J. Math. Phys. 24, 465-475 (2017).
- Taekyun Kim and Dae San Kim, Some identities of Catalan-Daehee polynomials arising from umbral calculus, Appl. Comput. Math. 16 (2017), no. 2, 177-189.
- Yuankui Ma, Taekyun Kim, Dae San Kim and Hyunseok Lee, Study on q-analogues of Catalan-Daehee numbers and polynomials, arXiv:2105.12013 [math.NT], 2021.
Crossrefs
Programs
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Mathematica
nmax:=36; a[n_]:=Denominator[Coefficient[Series[Log[1-4x]/(2(Sqrt[1-4x]-1)),{x,0,nmax}],x,n]]; Array[a,nmax,0] (* or *) a[n_]:=Denominator[If[n==0,1,4^n/(n+1)-Sum[4^(n-m-1)CatalanNumber[m]/(n-m),{m,0,n-1}]]]; Array[a,36,0]
Formula
G.f. of d(n): log(1 - 4*x)/(2*(sqrt(1 - 4*x) - 1)).
a(n) = denominator(d(n)), where d(n) = 4^n/(n + 1) - Sum_{m=0..n-1} 4^(n-m-1)*C(m)/(n - m) with d(0) = 1 and C(m) the m-th Catalan number.