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Central terms of triangles A102472 and A102473.
3

%I #25 Jul 12 2024 11:27:23

%S 1,2,13,130,1807,32280,705421,18237164,544505521,18438430990,

%T 698246022001,29239344782022,1341545985079903,66926098621724300,

%U 3606825675219961657,208826700420103831480,12926842112341879416001,851962999949978920707834,59561112879709434549509941

%N Central terms of triangles A102472 and A102473.

%H Reinhard Zumkeller, <a href="/A247365/b247365.txt">Table of n, a(n) for n = 1..300</a>

%F a(n) = A102472(2*n-1,n) = A102473(2*n-1,n).

%F 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 a(n) = round(2*BesselI(n-1,2)*BesselK(2*n-1,2)). - _Mark van Hoeij_, Nov 08 2022

%F a(n) ~ 2^(2*n - 3/2) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Nov 09 2022

%F 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 seq(round(2*BesselI(n-1,2)*BesselK(2*n-1,2)), n=1..30); # _Mark van Hoeij_, Nov 08 2022

%p A001040 := proc(n) options remember;

%p if n < 2 then n else (n - 1)*procname(n-1) + procname(n-2) fi

%p end:

%p A001053 := proc(n) options remember;

%p if n < 2 then 1-n else (n - 1)*procname(n-1) + procname(n-2) fi

%p end:

%p 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 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 (Haskell)

%o a247365 n = a102473 (2 * n - 1) n

%Y Cf. A102472, A102473, A058294.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Sep 14 2014