OFFSET
0,3
LINKS
Winston de Greef, Table of n, a(n) for n = 0..438
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 = 2.
a(n) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(n+m-1,n-i)*(n-i)!*m^(i-1) where m = 2 for n >= 1.
E.g.f.: (1+x^2)*exp(2/(1-x))/(4*(1-x)^2*exp(2))+3/4. - Alois P. Heinz, Mar 19 2023
a(n) ~ 2^(-9/4) * exp(2*sqrt(2*n) - n - 1) * n^(n + 3/4). - Vaclav Kotesovec, Mar 20 2023
MAPLE
# For recursion:
N:=20; a[0]:=1; a[1]:=1; for n from 1 to N do
a[n+1]:=(n+2)*(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+1, n-i)*(n-i)!*2^(i-1), i = 0..n-1)):
seq(a(n), n = 0..20);
PROG
(PARI)
memo=Map([0, 1; 1, 1]);
a(n)=if(mapisdefined(memo, n), mapget(memo, n), mapput(memo, n, (n+1)* (2*a(n-1) - (n-2)*a(n-2))); a(n)); \\ Winston de Greef, Mar 20 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Igor Victorovich Statsenko, Mar 19 2023
STATUS
approved