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A361649
a(n) = (1+n)*(2*a(n-1) - (n-2)*a(n-2)) with a(0) = a(1) = 1.
3
1, 1, 6, 44, 380, 3768, 42112, 523072, 7141248, 106209920, 1708188416, 29525850624, 545607622144, 10730032423936, 223691600732160, 4926284479250432, 114255071320260608, 2783085758131765248, 71023717127647854592, 1894699527341113999360, 52730415074075898937344
OFFSET
0,3
LINKS
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
For m=1 the formula gives the sequence A052852.
Sequence in context: A147688 A090442 A286867 * A084965 A203159 A279085
KEYWORD
nonn
AUTHOR
STATUS
approved