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A118353 Semi-diagonal (two rows below central terms) of pendular triangle A118350 and equal to the self-convolution cube of the central terms (A118351).

%I A118353 #10 Feb 19 2021 09:45:27
%S A118353 1,3,21,163,1353,11760,105681,973953,9154821,87428388,845894700,
%T A118353 8273978100,81682757317,812829371205,8144563709391,82104333340467,
%U A118353 832125695906313,8473862660311392,86661931504395228,889705959333345756
%N A118353 Semi-diagonal (two rows below central terms) of pendular triangle A118350 and equal to the self-convolution cube of the central terms (A118351).
%H A118353 G. C. Greubel, <a href="/A118353/b118353.txt">Table of n, a(n) for n = 0..500</a>
%t A118353 T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==n, 0, T[n-1, k] - 3*T[n-1, k-1] + 3*T[n, k-1] + T[n+1, k-1] ]];
%t A118353 Table[T[n, n-3], {n, 3, 30}] (* _G. C. Greubel_, Feb 18 2021 *)
%o A118353 (PARI) my(x='x+O('x^33)); Vec((serreverse(x*(1-3*x+sqrt((1-3*x)*(1-7*x)))/2/(1-3*x))/x)^3)
%o A118353 (Sage)
%o A118353 @CachedFunction
%o A118353 def T(n, k):
%o A118353     if (k<0 or n<k): return 0
%o A118353     elif (k==0): return 1
%o A118353     elif (k==n): return 0
%o A118353     else: return T(n-1, k) - 3*T(n-1, k-1) + 3*T(n, k-1) + T(n+1, k-1)
%o A118353 [T(n,n-3) for n in (3..30)] # _G. C. Greubel_, Feb 18 2021
%Y A118353 Cf. A118350, A118351, A118352, A118354.
%K A118353 nonn
%O A118353 0,2
%A A118353 _Paul D. Hanna_, Apr 26 2006