A344849 a(n) is the numerator of Catalan-Daehee number d(n).
1, 1, 7, 20, 313, 344, 24634, 86008, 183349, 3301264, 132174038, 69326344, 3332927794, 17361255440, 108222173516, 406589577424, 26070625295573, 8970328188896, 55462481190898, 1055714050810664, 2169454884422962, 91277283963562352, 8046203518285051612, 686567135431420560
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:=24; a[n_]:=Numerator[Coefficient[Series[Log[1-4x]/(2(Sqrt[1-4x]-1)),{x,0,nmax}],x,n]]; Array[a,nmax,0] (* or *) a[n_]:=Numerator[If[n==0,1,4^n/(n+1)-Sum[4^(n-m-1)CatalanNumber[m]/(n-m),{m,0,n-1}]]]; Array[a,24,0]
Formula
G.f. of d(n): log(1 - 4*x)/(2*(sqrt(1 - 4*x) - 1)).
a(n) = numerator(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) is the m-th Catalan number.