A064571 Binomial transform of (3n)!.

1, 7, 733, 365059, 480457465, 1310073012031, 6410226944228437, 51135786265589035963, 620857308610340190325489, 10894455325845833715898309495, 265361776433005578042408841530061, 8686235999260031219427964764144394867, 372097544110411566152076374938114937481193

Offset: 0

Author

Karol A. Penson, Sep 20 2001

Keywords

• nonn
• easy

Comments

From Peter Bala, Mar 20 2022: (Start)

The congruence a(n+k) == a(n) (mod k) holds for all n and k.

It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, taken modulo 5 the sequence becomes [2, 3, 4, 0, 1, 2, 3, 4, 0, 1, ...], a purely periodic sequence with period 5.

More generally, the same property holds for any sequence with an e.g.f. of the form F(x)*exp(x*G(x)), where F(x) and G(x) are power series with integer coefficients and G(0) = 1 (see Bala, Theorem 1). (End)

Links

• Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Crossrefs

Cf. A000522, A064570.

Programs

Mathematica

Table[Sum[Binomial[n, k] * (3*k)!, {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 30 2017 *)

PARI

for(n=0,12,print1(round(intnum(x=0,999,exp(-x)*(1+x^3)^n)),", ")) - Gerald McGarvey, Oct 12 2007

Formula

In Maple notation: a(n) = hypergeom([1, 1/3, 2/3, -n], [], -27), n = 0, 1, ....

a(n) = Integral_{x = 0..infinity} (1+x^3)^n*exp(-x) dx. - Gerald McGarvey, Oct 12 2007

From Vaclav Kotesovec, Oct 30 2017: (Start)

a(n) = (27*n^3 - 27*n^2 + 6*n + 1)*a(n-1) - 3*(n-1)*(27*n^2 - 45*n + 20)*a(n-2) + 27*(n-2)*(n-1)*(3*n - 4)*a(n-3) - 27*(n-3)*(n-2)*(n-1)*a(n-4).

a(n) ~ sqrt(2*Pi) * (3*n)^(3*n + 1/2) / exp(3*n). (End)

From Peter Bala, Mar 20 2022: (Start)

a(n) = Sum_{k = 0..n} binomial(n,k)*(3*k)!.

a(n) = 1 + 6*n*(1 + 60*(n-1)*(1 + 168*(n-2)*(1 + 330*(n-3)*(1 + 546*(n-4)*(1 + ...*(1 + (3*(3*m+1)*(3*m+2))*(n-m)*(1 + ...))))))).

a(n) == 1 (mod 6).

O.g.f: Sum_{k >= 0} (3*k)!*x^k/(1 - x)^(k+1).

E.g.f.: exp(x)*Sum_{k >= 0} (3*k)!/k!*x^k. (End)

Extensions

Corrected and extended by N. J. A. Sloane, Oct 29 2006