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

Author

Sarah Arpin, Apr 29 2019

Keywords

• sign

Examples

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

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.

Crossrefs

Inverse binomial transform of A003418 (shifted).

Programs

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 *)

PARI

a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*lcm(vector(k+1, i, i))); \\ Michel Marcus, Apr 30 2019

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)

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).