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

0,2

Comments

Inverse binomial transform gives the factorial numbers up to sign.

Links

Table of n, a(n) for n=0..23.

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.

Example

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}
.

Prog

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

Crossrefs

Cf. A000522, A182386.

Sequence in context: A286381 A286382 A127580 * A098048 A257348 A101847

Adjacent sequences: A382179 A382180 A382181 * A382183 A382184 A382185

Keyword

nonn

Author

Thomas Scheuerle, Mar 17 2025

Status

approved