A321849 Expansion of e.g.f.: exp(x/(1-6*x)).

1, 1, 13, 253, 6553, 211801, 8201701, 369979093, 19047250993, 1101705494833, 70715424362941, 4987040544656941, 383243311962126793, 31871863566298601353, 2851588139929576342933, 273093945561709965890821, 27871997808801906673665121

Offset: 0

Author

Ludovic Schwob, Nov 19 2018

Keywords

• nonn

Comments

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Links

• Ludovic Schwob, Table of n, a(n) for n = 0..199

Crossrefs

Cf. A000262, A025168, A321837, A321847, A321848, A321850.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-6*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 08 2018

Magma

[1] cat [&+[6^(n-k)* Factorial(n)/Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018

Maple

seq(coeff(series(factorial(n)*exp(x/(1-6*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

Mathematica

a[n_] := Sum[6^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (12n - 11)*a[n - 1] - 36(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

PARI

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-6*x)))) \\ Michel Marcus, Nov 25 2018

Sage

f= exp(x/(1-6*x))
g=f.taylor(x,0,13)
L=g.coefficients()
coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
coeffs  # G. C. Greubel, Dec 08 2018

Formula

a(n) = Sum_{k=0..n} 6^(n-k)*(n!/k!)*binomial(n-1, k-1).

Recurrence: a(n) = (12*n-11)*a(n-1) - 36*(n-2)*(n-1)*a(n-2).

a(n) ~ n! * exp(sqrt(2*n/3) - 1/12) * 6^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018