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

0,3

Links

Table of n, a(n) for n=0..23.

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

Cf. A000108, A000302, A014973 (denominators of Daehee numbers), A343206, A344850 (denominators).

Sequence in context: A299800 A299676 A300308 * A216457 A266048 A233331

Adjacent sequences: A344846 A344847 A344848 * A344850 A344851 A344852

Keyword

nonn,frac

Author

Stefano Spezia, May 30 2021

Status

approved