A118353 - OEIS

%I #10 Feb 19 2021 09:45:27

%S 1,3,21,163,1353,11760,105681,973953,9154821,87428388,845894700,

%T 8273978100,81682757317,812829371205,8144563709391,82104333340467,

%U 832125695906313,8473862660311392,86661931504395228,889705959333345756

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

%H G. C. Greubel, <a href="/A118353/b118353.txt">Table of n, a(n) for n = 0..500</a>

%t 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 Table[T[n, n-3], {n, 3, 30}] (* _G. C. Greubel_, Feb 18 2021 *)

%o (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 (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (k<0 or n<k): return 0

%o     elif (k==0): return 1

%o     elif (k==n): return 0

%o     else: return T(n-1, k) - 3*T(n-1, k-1) + 3*T(n, k-1) + T(n+1, k-1)

%o [T(n,n-3) for n in (3..30)] # _G. C. Greubel_, Feb 18 2021

%Y Cf. A118350, A118351, A118352, A118354.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 26 2006