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.

1, 2, 5, 16, 17, 86, 1237, 1940, 25601, 617482, 1386821, 25329272, 815052625, 2379750686, 55319082197, 2225093600956, 7995962217857, 225701855249810, 10894058270134021, 46488524334434912, 1543800689908468241, 86934584995669200742, 429553964850178236245, 16404426130967383104356

Offset: 0

Author

Thomas Scheuerle, Mar 17 2025

Keywords

• nonn

Comments

Inverse binomial transform gives the factorial numbers up to sign.

Examples

			The first 6 polynomials are:
 1 = {1} for x = {0}
 x + 1 = {1, 2} for x = {0, 1}
 x^2 + 1 = {1, 2} for x = {0, 1, 2}
 x^3 - 2*x^2 + 2*x + 1  = {1, 2, 5} for x = {0, 1, 2, 3}
-x^4 + 7*x^3 - 13*x^2 + 8*x + 1  = {1, 2, 5, 16} for x = {0, 1, 2, 3, 4}
 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}
.
		

Crossrefs

Cf. A000522, A182386.

Programs

PARI

a(n) = sum(m=1, n+1, binomial(n, m-1)*(m-1)!*(1-2*(m%3==2 && m>3)))

Formula

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.

If k = a(n) then k divides a(n+m*k) for some m.