A279127 a(n) = Sum_{0<=m

0, 1, 8, 147, 5824, 405845, 43733976, 6726601063, 1398047697152, 377278848390249, 128228860181918440, 53585748788874537851, 27001973543813627400768, 16144773936121968789213757, 11300021011239061076228900024, 9152162639827097780662174019535

Offset: 0

Author

Robert Israel, Dec 06 2016

Keywords

• nonn

Comments

n-m | a(n)-a(m) for all n,m.

Links

• Robert Israel, Table of n, a(n) for n = 0..225
• Math StackExchange, Are certain integer functions well-defined modulo different primes necessarily polynomials?

Crossrefs

Cf. A003470.

Programs

Maple

f:= n -> add(mul(n-m,m=-k..k),k=0..n):
map(f, [$0..40]);

Mathematica

Table[Sum[Product[n - j, {j, -m, m}], {m,0,n}], {n,0,25}] (* G. C. Greubel, Dec 07 2016 *)

PARI

a(n) = sum(m=0, n-1, prod(j=-m, m, n-j)); \\ Michel Marcus, Dec 07 2016

Formula

a(n) = A003470(2n-1) for n >= 1.

a(n) = n*hypergeom([1,n+1,1-n],[],-1).

a(n+3) = -a(n)+(4*n^2+6*n-1)*a(n+1)+(4*n^2+18*n+17)*a(n+2)+8*n+12.

D-finite with recurrence +(-2*n+5)*a(n) +(2*n-5)*(4*n^2-6*n+1)*a(n-1) -(2*n-1)*(4*n^2-18*n+19)*a(n-2) +(2*n-1)*a(n-3)=0. - R. J. Mathar, Jul 27 2022