A352681 a(n) = [x^n] (3*x + 1)*(1 - sqrt(1 - 4*x))/(2*x).

1, 4, 5, 11, 29, 84, 258, 825, 2717, 9152, 31382, 109174, 384370, 1366936, 4903140, 17718165, 64442205, 235717800, 866573070, 3200179290, 11865909990, 44158628280, 164881364700, 617507304570, 2319082988274, 8731658843424, 32953192276508, 124635610132460

Offset

0,2

Links

Table of n, a(n) for n=0..27.

Formula

a(n) = A352680(4, n).

D-finite with recurrence a(n) = a(n-1)*((14*n + 2)*(2*n - 3))/((7*n - 6)*(n + 1)) for n >= 2.

a(n) = A000108(n)+3*A000108(n-1). - R. J. Mathar, Mar 29 2022

a(n) ~ 7 * 4^(n-1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 24 2025

Maple

alias(PS = ListTools:-PartialSums):
aList := proc(len) local a, k, P, T; a := 4; P := [1]; T := [1];
for k from 0 to len-1 do T := [op(T), a]; P := PS([op(P), a]); a := P[-1] od;
T end: aList(27);
# Alternative:
a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 4 else
a(n-1)*((14*n + 2)*(2*n - 3))/((7*n - 6)*(n + 1)) fi end: seq(a(n), n = 0..27);

Mathematica

a[n_] := CatalanNumber[n] + 3 * CatalanNumber[n-1]; a[0] = 1; Array[a, 30, 0] (* Amiram Eldar, Sep 24 2025 *)

Crossrefs

Cf. A000108, A352680.

Sequence in context: A056799 A216963 A259821 * A109503 A304224 A304135

Adjacent sequences: A352678 A352679 A352680 * A352682 A352683 A352684

Keyword

nonn,easy

Author

Peter Luschny, Mar 27 2022

Status

approved