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A247365 Central terms of triangles A102472 and A102473.

%I A247365 #25 Jul 12 2024 11:27:23
%S A247365 1,2,13,130,1807,32280,705421,18237164,544505521,18438430990,
%T A247365 698246022001,29239344782022,1341545985079903,66926098621724300,
%U A247365 3606825675219961657,208826700420103831480,12926842112341879416001,851962999949978920707834,59561112879709434549509941
%N A247365 Central terms of triangles A102472 and A102473.
%H A247365 Reinhard Zumkeller, <a href="/A247365/b247365.txt">Table of n, a(n) for n = 1..300</a>
%F A247365 a(n) = A102472(2*n-1,n) = A102473(2*n-1,n).
%F A247365 a(n) = y(n,n), where y(m+2,n) = (m + n)*y(m+1,n) + y(m,n), with y(0,n)=0, y(1,n)=1 for all n. - _Benedict W. J. Irwin_, Nov 03 2016
%F A247365 a(n) = round(2*BesselI(n-1,2)*BesselK(2*n-1,2)). - _Mark van Hoeij_, Nov 08 2022
%F A247365 a(n) ~ 2^(2*n - 3/2) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Nov 09 2022
%F A247365 a(n) = (-1)^n * (A001040(n-1) * A001053(2*n-1) - A001053(n-1) * A001040(2*n-1)). - _Mark van Hoeij_, Jul 10 2024
%p A247365 seq(round(2*BesselI(n-1,2)*BesselK(2*n-1,2)), n=1..30); # _Mark van Hoeij_, Nov 08 2022
%p A247365 A001040 := proc(n) options remember;
%p A247365   if n < 2 then n else (n - 1)*procname(n-1) + procname(n-2) fi
%p A247365 end:
%p A247365 A001053 := proc(n) options remember;
%p A247365   if n < 2 then 1-n else (n - 1)*procname(n-1) + procname(n-2) fi
%p A247365 end:
%p A247365 seq( (-1)^n * (A001040(n-1) * A001053(2*n-1) - A001053(n-1) * A001040(2*n-1)), n=1..30); # _Mark van Hoeij_, Jul 10 2024
%t A247365 Table[DifferenceRoot[Function[{y,m},{y[2+m]==(m+n)y[1+m]+y[m],y[0]==0,y[1]==1}]][n],{n,1,20}] (* _Benedict W. J. Irwin_, Nov 03 2016 *)
%o A247365 (Haskell)
%o A247365 a247365 n = a102473 (2 * n - 1) n
%Y A247365 Cf. A102472, A102473, A058294.
%K A247365 nonn
%O A247365 1,2
%A A247365 _Reinhard Zumkeller_, Sep 14 2014