A181088 - OEIS

%I #11 Apr 05 2021 00:07:06

%S 1,-4,-40,672,8064,-253440,-3294720,153753600,2091048960,

%T -130025226240,-1820353167360,141707492720640,2024392753152000,

%U -189483161695027200,-2747505844577894400,300609462994993152000,4408938790593232896000

%N a(n) = A181089(2*n+1,n)/(n+2).

%C What are the constraints on left-right symmetric triangles t(n,m) such that t(2*n,n)/(n+1) are integers?

%H G. C. Greubel, <a href="/A181088/b181088.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = (A060821(2*n+1, n) + A060821(2*n+1, n+1))/(n+2). - _G. C. Greubel_, Apr 04 2021

%t (* First program *)

%t p[x_, n_] = HermiteH[n, x] + ExpandAll[x^n*HermiteH[n, 1/x]];

%t b:= Table[CoefficientList[p[x, n], x], {n, 0, 50}];

%t Table[b[[2*n+2, n+1]]/(n+2), {n,0,20}]

%t (* Second program *)

%t A060821[n_, k_]:= If[EvenQ[n-k], (-1)^(Floor[(n-k)/2])*2^k*n!/(k!*(Floor[(n - k)/2]!)), 0];

%t a[n_]:= (A060821[2*n+1, n] + A060821[2*n+1, n+1])/(n+2);

%t Table[a[n], {n, 0, 25}] (* _G. C. Greubel_, Apr 04 2021 *)

%o (Sage)

%o def A060821(n,k): return (-1)^((n-k)//2)*2^k*factorial(n)/(factorial(k)*factorial( (n-k)//2)) if (n-k)%2==0 else 0

%o def a(n): return (A060821(2*n+1, n) + A060821(2*n+1, n+1))/(n+2)

%o [a(n) for n in (0..25)] # _G. C. Greubel_, Apr 04 2021

%Y Cf. A060821, A177042, A177043, A181089.

%K sign

%O 0,2

%A _Roger L. Bagula_, Oct 02 2010