OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
I. V. Statsenko, Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences, Applied Discrete Mathematics No. 55, Tomsk State University Publishing House, 2022, pp. 5-13.
FORMULA
a(n) = (m+n-1)*(2*a(n-1) - (n-2)*a(n-2)) where m=3, a(0)=a(1)=1.
a(n) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(n+m-1,n-i)*(n-i)!*m^(i-1) where m = 3 for n >= 1.
a(n) = (n + 2)!*hypergeom([1 - n], [3], -3) / 6) for n >= 1. - Peter Luschny, Mar 23 2023
From Vaclav Kotesovec, Mar 23 2023: (Start)
E.g.f.: 23/27 + (4 + 3*x + 2*x^3) * exp(3*x/(1-x)) / (27*(1-x)^3).
a(n) ~ exp(2*sqrt(3*n) - n - 3/2) * n^(n + 5/4) / (sqrt(2) * 3^(9/4)). (End)
MAPLE
# For recursion:
N:=10; a[0]:=1; a[1]:=1; for n from 1 to N do
a[n+1]:=(n+3)*(2*a[n]-(n-1)*a[n-1]); od;
# For closed form:
C := binomial:
a := n -> `if`(n=0, 1, add(C(n-1, i)*C(n+2, n-i)*(n-i)!*3^(i-1), i = 0..n-1)):
seq(a(n), n = 0..20);
# Alternative:
a := n -> `if`(n=0, 1, (n + 2)!*hypergeom([1 - n], [3], -3) / 6):
seq(simplify(a(n)), n = 0..20); # Peter Luschny, Mar 23 2023
MATHEMATICA
nmax = 20; CoefficientList[Series[23/27 + (4 + 3*x + 2*x^3)*E^(3*x/(1 - x))/(27*(1 - x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 23 2023 *)
PROG
(PARI) a(n) = if(n==0, 1, my(m=3); sum(i=0, n-1, binomial(n-1, i)*binomial(n+m-1, n-i)*(n-i)!*m^(i-1))) \\ Andrew Howroyd, Mar 23 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Igor Victorovich Statsenko, Mar 23 2023
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Mar 23 2023
STATUS
approved