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A307803 Inverse binomial transform of least common multiple sequence.

%I A307803 #23 Nov 27 2020 02:08:32
%S A307803 1,-1,3,1,41,171,799,2633,7881,24391,99611,461649,2252953,10773491,
%T A307803 46602711,176413201,596116769,1899975183,6302881171,24136694081,
%U A307803 105765310281,476455493179,2033813426063,8019234229401,29410337173561,102444237073751,347418130583499
%N A307803 Inverse binomial transform of least common multiple sequence.
%H A307803 Alois P. Heinz, <a href="/A307803/b307803.txt">Table of n, a(n) for n = 0..1000</a>
%H A307803 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 A307803 a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*A003418(k+1).
%F A307803 Formula for values modulo 10: (Proof by considering the formula modulo 10)
%F A307803     a(n) (mod 10) = 1, if n = 0, 3, 4 (mod 5),
%F A307803     a(n) (mod 10) = 9, if n = 1 (mod 5),
%F A307803     a(n) (mod 10) = 3, if n = 2 (mod 5).
%e A307803 For n = 3, a(3) = binomial(3,0)*1 - binomial(3,1)*2 + binomial(3,2)*6 - binomial(3,3)*12 = 1.
%p A307803 b:= proc(n) option remember; `if`(n=0, 1, ilcm(n, b(n-1))) end:
%p A307803 a:= n-> add(b(i+1)*binomial(n, i)*(-1)^i, i=0..n):
%p A307803 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 29 2019
%t A307803 b[n_] := b[n] = If[n == 0, 1, LCM[n, b[n - 1]]];
%t A307803 a[n_] := Sum[b[i + 1] Binomial[n, i] (-1)^i, {i, 0, n}];
%t A307803 a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)
%o A307803 (Sage)
%o A307803 def SIbinomial_transform(N, seq):
%o A307803     BT = [seq[0]]
%o A307803     k = 1
%o A307803     while k< N:
%o A307803         next = 0
%o A307803         j = 0
%o A307803         while j <=k:
%o A307803             next = next + (((-1)^j)*(binomial(k,j))*seq[j])
%o A307803             j = j+1
%o A307803         BT.append(next)
%o A307803         k = k+1
%o A307803     return BT
%o A307803 LCMSeq = []
%o A307803 for k in range(1,26):
%o A307803     LCMSeq.append(lcm(range(1,k+1)))
%o A307803 SIbinomial_transform(25, LCMSeq)
%o A307803 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*lcm(vector(k+1, i, i))); \\ _Michel Marcus_, Apr 30 2019
%Y A307803 Inverse binomial transform of A003418 (shifted).
%K A307803 sign
%O A307803 0,3
%A A307803 _Sarah Arpin_, Apr 29 2019