OFFSET
0,3
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.
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.
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]
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Stefano Spezia, May 30 2021
STATUS
approved