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a(n) = Sum_{i+j+k=n, 0<=j<=i<=n, 0<=k<=n} (n+k)!/(i! * j! * (2*k)!).
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%I #15 Oct 30 2021 05:29:25

%S 1,2,7,28,104,415,1631,6438,25557,101320,402620,1600973,6369850,

%T 25362023,101021833,402558824,1604694342,6398518221,25519847999,

%U 101805661146,406209454697,1621073551580,6470257616586,25828532256195

%N a(n) = Sum_{i+j+k=n, 0<=j<=i<=n, 0<=k<=n} (n+k)!/(i! * j! * (2*k)!).

%C for n>=1 a(n) mod 2 = A059448(n) the parity of number of 0's in binary expansion of n.

%H Seiichi Manyama, <a href="/A092465/b092465.txt">Table of n, a(n) for n = 0..200</a>

%F From _Vaclav Kotesovec_, Oct 30 2021: (Start)

%F Recurrence: n*(2*n - 1)*(1024*n^4 - 10688*n^3 + 40544*n^2 - 65368*n + 37617)*a(n) = (14336*n^6 - 169088*n^5 + 773760*n^4 - 1725328*n^3 + 1933390*n^2 - 999891*n + 175950)*a(n-1) - (14336*n^6 - 189568*n^5 + 983936*n^4 - 2540080*n^3 + 3358598*n^2 - 2079295*n + 445620)*a(n-2) - (38912*n^6 - 503424*n^5 + 2607232*n^4 - 6826768*n^3 + 9326358*n^2 - 6043703*n + 1296960)*a(n-3) - 2*(28672*n^6 - 362752*n^5 + 1804800*n^4 - 4416032*n^3 + 5425772*n^2 - 2994798*n + 432585)*a(n-4) + 8*(28672*n^6 - 403712*n^5 + 2232064*n^4 - 6111200*n^3 + 8528476*n^2 - 5498774*n + 1180095)*a(n-5) - 64*(n-5)*(2*n - 7)*(1024*n^4 - 6592*n^3 + 14624*n^2 - 12248*n + 3129)*a(n-6).

%F a(n) ~ 4^n / 3 * (1 + sqrt(3)/(2*sqrt(Pi*n))). (End)

%o (PARI) a(n)=sum(i=0,n,sum(j=0,i,sum(k=0,n,if(i+j+k-n,0,(n+k)!/i!/j!/(2*k)!))))

%Y Cf. A092466.

%K nonn

%O 0,2

%A _Benoit Cloitre_, Mar 25 2004