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

%I A361649 #41 Feb 25 2024 10:08:19
%S A361649 1,1,6,44,380,3768,42112,523072,7141248,106209920,1708188416,
%T A361649 29525850624,545607622144,10730032423936,223691600732160,
%U A361649 4926284479250432,114255071320260608,2783085758131765248,71023717127647854592,1894699527341113999360,52730415074075898937344
%N A361649 a(n) = (1+n)*(2*a(n-1) - (n-2)*a(n-2)) with a(0) = a(1) = 1.
%H A361649 Winston de Greef, <a href="/A361649/b361649.txt">Table of n, a(n) for n = 0..438</a>
%H A361649 I. V. Statsenko, <a href="https://www.mathnet.ru/eng/pdm757">Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences</a>, Applied Discrete Mathematics No. 55, Tomsk State University Publishing House, 2022, pp. 5-13.
%F A361649 a(n) = (m+n-1)*(2*a(n-1) - (n-2)*a(n-2)) where m = 2.
%F A361649 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.
%F A361649 E.g.f.: (1+x^2)*exp(2/(1-x))/(4*(1-x)^2*exp(2))+3/4. - _Alois P. Heinz_, Mar 19 2023
%F A361649 a(n) ~ 2^(-9/4) * exp(2*sqrt(2*n) - n - 1) * n^(n + 3/4). - _Vaclav Kotesovec_, Mar 20 2023
%p A361649 # For recursion:
%p A361649 N:=20;a[0]:=1;a[1]:=1;for n from 1 to N do
%p A361649 a[n+1]:=(n+2)*(2*a[n]-(n-1)*a[n-1]);od;
%p A361649 # For closed form:
%p A361649 C := binomial:
%p A361649 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)):
%p A361649 seq(a(n), n = 0..20);
%o A361649 (PARI)
%o A361649 memo=Map([0, 1; 1, 1]);
%o A361649 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
%Y A361649 For m=1 the formula gives the sequence A052852.
%Y A361649 Cf. A288268, A361528.
%K A361649 nonn
%O A361649 0,3
%A A361649 _Igor Victorovich Statsenko_, Mar 19 2023