A003470 -id:A003470 - cp's OEIS frontend

A358494 a(n) = Sum_{k=0..floor(n/5)} (n-4*k)!/k!.

1, 1, 2, 6, 24, 121, 722, 5046, 40344, 363000, 3629521, 39921843, 479041932, 6227383740, 87181920360, 1307714287321, 20923268909764, 355693655298260, 6402460885833720, 121646408103159240, 2432922931206035521, 51091297862251106885, 1124007130194903158430

Offset: 0

Seiichi Manyama, Nov 19 2022

Cf. A000522, A003470, A357949, A358493.

PARI
```
a(n) = sum(k=0, n\5, (n-4*k)!/k!);
```

a(n) = (n-1) * a(n-1) + (n-2) * a(n-2) + (n-3) * a(n-3) + (n-4) * a(n-4) + (n-8) * a(n-5) - 4 * a(n-6) - 4 * a(n-7) - 4 * a(n-8) - 4 * a(n-9) + 5 for n > 8.

G.f.: Sum_{k>=0} k! * x^k/(1-x^5)^(k+1). - Seiichi Manyama, Feb 26 2024

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

Original entry on oeis.org

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

Offset: 0

Robert Israel, Dec 06 2016

nonn

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

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

Cf. A003470.

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

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

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

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

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A358494 a(n) = Sum_{k=0..floor(n/5)} (n-4*k)!/k!.

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

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