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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
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
KEYWORD
nonn
AUTHOR
Sarah Arpin, Apr 29 2019
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)