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A307800 Binomial transform of least common multiple sequence (A003418), starting with a(1). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

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} 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).

EXAMPLE

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

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 *)

PROG

(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)

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

CROSSREFS

Binomial transform of A003418 (shifted).

Sequence in context: A047102 A109000 A183511 * A265796 A129962 A026361

Adjacent sequences:  A307797 A307798 A307799 * A307801 A307802 A307803

KEYWORD

nonn

AUTHOR

Sarah Arpin, Apr 29 2019

STATUS

approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)