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A307800
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Binomial transform of least common multiple sequence (A003418), starting with a(1).
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0
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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
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OFFSET
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0,2
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LINKS
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Jackson Earles, Aaron Li, Adam Nelson, Marlo Terr, Sarah Arpin, and Ilia Mishev Binomial Transforms of Sequences, CU Boulder Experimental Math Lab, Spring 2019.
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FORMULA
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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).
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EXAMPLE
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For n = 3, a(3) = binomial(3,0)*1 + binomial(3,1)*2 + binomial(3,2)*6 + binomial(3,3)*12 = 37.
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MAPLE
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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):
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MATHEMATICA
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Table[Sum[Binomial[n, k]*Apply[LCM, Range[k+1]], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jun 06 2019 *)
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PROG
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(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
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CROSSREFS
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Binomial transform of A003418 (shifted).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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