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

0,3

Links

Table of n, a(n) for n=0..35.

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) = 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.

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]

Crossrefs

Cf. A000108, A000302, A014973 (denominators of Daehee numbers), A343206, A344849 (numerators).

Sequence in context: A179857 A260078 A163590 * A114320 A185138 A285947

Adjacent sequences: A344847 A344848 A344849 * A344851 A344852 A344853

Keyword

nonn,frac

Author

Stefano Spezia, May 30 2021

Status

approved