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A307800 Binomial transform of least common multiple sequence (A003418), starting with a(1). 0

%I

%S 1,3,11,37,153,551,2023,7701,29417,107083,384771,1408133,5457961,

%T 22466367,92977823,365613181,1342359393,4677908531,16159185307,

%U 58676063493,231520762361,967464685783,4052593703511,16354948948517,62709285045913,229276436653851

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

%H Jackson Earles, Aaron Li, Adam Nelson, Marlo Terr, Sarah Arpin, and Ilia Mishev <a href="https://www.colorado.edu/math/binomial-transforms-sequences-spring-2019">Binomial Transforms of Sequences</a>, CU Boulder Experimental Math Lab, Spring 2019.

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

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

%F a(n) (mod 10) = 1, if n = 0, 2 (mod 5),

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

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

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

%p b:= proc(n) option remember; `if`(n=0, 1, ilcm(n, b(n-1))) end:

%p a:= n-> add(b(i+1)*binomial(n, i), i=0..n):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 29 2019

%t Table[Sum[Binomial[n, k]*Apply[LCM, Range[k+1]], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Jun 06 2019 *)

%o (Sage)

%o def OEISbinomial_transform(N, seq):

%o BT = [seq[0]]

%o k = 1

%o while k< N:

%o next = 0

%o j = 0

%o while j <=k:

%o next = next + ((binomial(k,j))*seq[j])

%o j = j+1

%o BT.append(next)

%o k = k+1

%o return BT

%o LCMSeq = []

%o for k in range(1,26):

%o LCMSeq.append(lcm(range(1,k+1)))

%o OEISbinomial_transform(25, LCMSeq)

%o (PARI) a(n) = sum(k=0, n, binomial(n, k)*lcm(vector(k+1, i, i))); \\ _Michel Marcus_, Apr 30 2019

%Y Binomial transform of A003418 (shifted).

%K nonn

%O 0,2

%A _Sarah Arpin_, Apr 29 2019

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Last modified May 25 17:53 EDT 2020. Contains 334595 sequences. (Running on oeis4.)