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Semi-diagonal (one row below central terms) of pendular triangle A118350 and equal to the self-convolution of the central terms (A118351).
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%I #7 Feb 18 2021 14:18:15

%S 1,2,13,96,770,6530,57612,523446,4864795,46032288,441981816,

%T 4295393886,42172388820,417668676206,4167719552099,41861139949200,

%U 422890327921650,4294027462637528,43801007565527184,448625344231794792

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

%H G. C. Greubel, <a href="/A118352/b118352.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-2], {n,2,30}] (* _G. C. Greubel_, Feb 18 2021 *)

%o (PARI) {a(n)=polcoeff((serreverse(x*(1-3*x+sqrt((1-3*x)*(1-7*x)+x*O(x^n)))/2/(1-3*x))/x)^2,n)}

%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-2) for n in (2..30)] # _G. C. Greubel_, Feb 18 2021

%Y Cf. A118350, A118351, A118353, A118354.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 26 2006