A358451 Inverse Euler transform of the Riordan numbers, (A005043).

1, 0, 1, 1, 2, 5, 11, 28, 68, 174, 445, 1166, 3068, 8190, 21994, 59585, 162360, 445145, 1226376, 3394654, 9434260, 26317865, 73661588, 206809307, 582255448, 1643536725, 4650250254, 13186484316, 37468566744, 106666821221, 304200399505, 868977304140, 2486163857424

Offset

0,5

Links

Table of n, a(n) for n=0..32.

OEIS Wiki, Euler transform

Formula

a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 15 2024

Maple

EulerInvTransform := proc(f) local c, b;
    c := proc(n) option remember;
    ifelse(n = 0, f(0), f(n) - b(n, n-1)) end:
    b := proc(n, k) option remember;
    if n = 0 then return 1 elif k < 1 then return 0 fi;
    add(binomial(c(k) + j - 1, j)*b(n-k*j, k-1), j=0..n/k) end:
c end:
a := EulerInvTransform(A005043): seq(a(n), n = 0..32);

Mathematica

EulerInvTransform[seq_List] := Module[{final = {}}, Do[AppendTo[final, i*seq[[i]] - Sum[final[[d]]*seq[[i-d]], {d, i-1}]], {i, Length[seq]}]; Table[Sum[MoebiusMu[i/d]*final[[d]], {d, Divisors[i]}]/i, {i, Length[seq] }]];
A005043[n_] := A005043[n] = If[n <= 1, 1-n, (n-1)*(2*A005043[n-1] + 3*A005043[n-2])/(n+1)];
Join[{1}, EulerInvTransform[Array[A005043, 32]]] (* Jean-François Alcover, Jun 15 2024 *)

Prog

(SageMath)
z = PowerSeriesRing(ZZ, 'z').gen().O(33)
g = 1 + z + sqrt(1 - 2*z - 3*z**2)
f = -z * g.derivative() / g
print([1] + [sum(moebius(n // d) * f[d]
      for d in divisors(n)) // n for n in range(1, 33)])
(Python)
from typing import Callable
from functools import cache
from math import comb
# Define 'binomial' for compatibility with Maple.
def binomial(n: int, k: int) -> int:
    if 0 <= k <= n: return comb(n, k)
    if k <= n <  0: return comb(-k-1, n-k)*(-1)**(n-k)
    if n <  0 <= k: return comb(-n+k-1, k)*(-1)**k
    return 0
def EulerInvTransform(f: Callable) -> Callable:
    @cache
    def h(n: int, k: int) -> int:
        if n == 0: return 1
        if k <  1: return 0
        return sum(binomial(b(k)+j-1, j) * h(n-k*j, k-1)
               for j in range(1 + n // k))
    @cache
    def b(n: int) -> int:
        if n == 0: return f(0)
        return f(n) - h(n, n - 1)
    return b
a = EulerInvTransform(A005043)
print([a(n) for n in range(33)])

Crossrefs

Cf. A005043.

Sequence in context: A325916 A257790 A174145 * A124016 A121398 A359181

Adjacent sequences: A358448 A358449 A358450 * A358452 A358453 A358454

Keyword

nonn

Author

Peter Luschny, Nov 20 2022

Status

approved