A307803 Inverse binomial transform of least common multiple sequence.

1, -1, 3, 1, 41, 171, 799, 2633, 7881, 24391, 99611, 461649, 2252953, 10773491, 46602711, 176413201, 596116769, 1899975183, 6302881171, 24136694081, 105765310281, 476455493179, 2033813426063, 8019234229401, 29410337173561, 102444237073751, 347418130583499

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Jackson Earles, Aaron Li, Adam Nelson, Marlo Terr, Sarah Arpin, and Ilia Mishev Binomial Transforms of Sequences, CU Boulder Experimental Math Lab, Spring 2019.

Formula

a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*A003418(k+1).

Formula for values modulo 10: (Proof by considering the formula modulo 10)

a(n) (mod 10) = 1, if n = 0, 3, 4 (mod 5),

a(n) (mod 10) = 9, if n = 1 (mod 5),

a(n) (mod 10) = 3, if n = 2 (mod 5).

Example

For n = 3, a(3) = binomial(3,0)*1 - binomial(3,1)*2 + binomial(3,2)*6 - binomial(3,3)*12 = 1.

Maple

b:= proc(n) option remember; `if`(n=0, 1, ilcm(n, b(n-1))) end:
a:= n-> add(b(i+1)*binomial(n, i)*(-1)^i, i=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 29 2019

Mathematica

b[n_] := b[n] = If[n == 0, 1, LCM[n, b[n - 1]]];
a[n_] := Sum[b[i + 1] Binomial[n, i] (-1)^i, {i, 0, n}];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

Prog

(Sage)
def SIbinomial_transform(N, seq):
    BT = [seq[0]]
    k = 1
    while k< N:
        next = 0
        j = 0
        while j <=k:
            next = next + (((-1)^j)*(binomial(k, j))*seq[j])
            j = j+1
        BT.append(next)
        k = k+1
    return BT
LCMSeq = []
for k in range(1, 26):
    LCMSeq.append(lcm(range(1, k+1)))
SIbinomial_transform(25, LCMSeq)
(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*lcm(vector(k+1, i, i))); \\ Michel Marcus, Apr 30 2019

Crossrefs

Inverse binomial transform of A003418 (shifted).

Sequence in context: A270132 A050817 A125082 * A356819 A362166 A136517

Adjacent sequences: A307800 A307801 A307802 * A307804 A307805 A307806

Keyword

sign

Author

Sarah Arpin, Apr 29 2019

Status

approved