A307800 Binomial transform of least common multiple sequence (A003418), starting with a(1).

1, 3, 11, 37, 153, 551, 2023, 7701, 29417, 107083, 384771, 1408133, 5457961, 22466367, 92977823, 365613181, 1342359393, 4677908531, 16159185307, 58676063493, 231520762361, 967464685783, 4052593703511, 16354948948517, 62709285045913, 229276436653851

Offset: 0

Author

Sarah Arpin, Apr 29 2019

Keywords

• nonn

Examples

			For n = 3, a(3) = binomial(3,0)*1 + binomial(3,1)*2 + binomial(3,2)*6 + binomial(3,3)*12 = 37.
		

Links

• 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

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), i=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 29 2019

Mathematica

Table[Sum[Binomial[n, k]*Apply[LCM, Range[k+1]], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jun 06 2019 *)

PARI

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

Sage

def OEISbinomial_transform(N, seq):
    BT = [seq[0]]
    k = 1
    while k< N:
        next = 0
        j = 0
        while j <=k:
            next = next + ((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)))
OEISbinomial_transform(25, LCMSeq)

Formula

a(n) = Sum_{k=0..n} 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, 2 (mod 5),

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

a(n) (mod 10) = 7, if n = 3 (mod 5).