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%I #23 Nov 27 2020 02:08:32
%S 1,-1,3,1,41,171,799,2633,7881,24391,99611,461649,2252953,10773491,
%T 46602711,176413201,596116769,1899975183,6302881171,24136694081,
%U 105765310281,476455493179,2033813426063,8019234229401,29410337173561,102444237073751,347418130583499
%N Inverse binomial transform of least common multiple sequence.
%H Alois P. Heinz, <a href="/A307803/b307803.txt">Table of n, a(n) for n = 0..1000</a>
%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} (-1)^k*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, 3, 4 (mod 5),
%F a(n) (mod 10) = 9, if n = 1 (mod 5),
%F a(n) (mod 10) = 3, if n = 2 (mod 5).
%e For n = 3, a(3) = binomial(3,0)*1 - binomial(3,1)*2 + binomial(3,2)*6 - binomial(3,3)*12 = 1.
%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)*(-1)^i, i=0..n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 29 2019
%t b[n_] := b[n] = If[n == 0, 1, LCM[n, b[n - 1]]];
%t a[n_] := Sum[b[i + 1] Binomial[n, i] (-1)^i, {i, 0, n}];
%t a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)
%o (Sage)
%o def SIbinomial_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 + (((-1)^j)*(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 SIbinomial_transform(25, LCMSeq)
%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*lcm(vector(k+1, i, i))); \\ _Michel Marcus_, Apr 30 2019
%Y Inverse binomial transform of A003418 (shifted).
%K sign
%O 0,3
%A _Sarah Arpin_, Apr 29 2019