A361649 a(n) = (1+n)*(2*a(n-1) - (n-2)*a(n-2)) with a(0) = a(1) = 1.

1, 1, 6, 44, 380, 3768, 42112, 523072, 7141248, 106209920, 1708188416, 29525850624, 545607622144, 10730032423936, 223691600732160, 4926284479250432, 114255071320260608, 2783085758131765248, 71023717127647854592, 1894699527341113999360, 52730415074075898937344

Offset: 0

Author

Igor Victorovich Statsenko, Mar 19 2023

Keywords

• nonn

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.

Crossrefs

For m=1 the formula gives the sequence A052852.

Cf. A288268, A361528.

Programs

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);

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

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