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A382182 Lexicographically earliest increasing sequence starting with a(0) = 1 such that the polynomial which interpolates the first k values has degree k-1 and only integer coefficients.

%I A382182 #20 Mar 19 2025 10:18:17
%S A382182 1,2,5,16,17,86,1237,1940,25601,617482,1386821,25329272,815052625,
%T A382182 2379750686,55319082197,2225093600956,7995962217857,225701855249810,
%U A382182 10894058270134021,46488524334434912,1543800689908468241,86934584995669200742,429553964850178236245,16404426130967383104356
%N A382182 Lexicographically earliest increasing sequence starting with a(0) = 1 such that the polynomial which interpolates the first k values has degree k-1 and only integer coefficients.
%C A382182 Inverse binomial transform gives the factorial numbers up to sign.
%F A382182 a(n) = Sum_{m=1..n+1} binomial(n, m-1)*(m-1)!*s(m), where s(m) = -1 if m > 3 and m == 2 (mod 3) in all other cases s(m) = 1.
%F A382182 If k = a(n) then k divides a(n+m*k) for some m.
%e A382182 The first 6 polynomials are:
%e A382182  1 = {1} for x = {0}
%e A382182  x + 1 = {1, 2} for x = {0, 1}
%e A382182  x^2 + 1 = {1, 2} for x = {0, 1, 2}
%e A382182  x^3 - 2*x^2 + 2*x + 1  = {1, 2, 5} for x = {0, 1, 2, 3}
%e A382182 -x^4 + 7*x^3 - 13*x^2 + 8*x + 1  = {1, 2, 5, 16} for x = {0, 1, 2, 3, 4}
%e A382182  x^5 - 11*x^4 + 42*x^3 - 63*x^2 + 32*x + 1  = {1, 2, 5, 16, 17} for x = {0, 1, 2, 3, 4, 5}
%e A382182 .
%o A382182 (PARI) a(n) = sum(m=1, n+1, binomial(n, m-1)*(m-1)!*(1-2*(m%3==2 && m>3)))
%Y A382182 Cf. A000522, A182386.
%K A382182 nonn
%O A382182 0,2
%A A382182 _Thomas Scheuerle_, Mar 17 2025