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A118354 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118351.

%I A118354 #10 Feb 18 2021 14:16:10
%S A118354 1,1,0,1,1,0,1,2,6,0,1,3,13,42,0,1,4,21,96,325,0,1,5,30,163,770,2688,
%T A118354 0,1,6,40,244,1353,6530,23286,0,1,7,51,340,2093,11760,57612,208659,0,
%U A118354 1,8,63,452,3010,18636,105681,523446,1918314,0,1,9,76,581,4125,27441,170580,973953,4864795,17994264,0
%N A118354 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118351.
%C A118354 A118351 equals the central terms of pendular triangle A118350 and the lower diagonals of this triangle form the semi-diagonals of the triangle A118350.
%H A118354 G. C. Greubel, <a href="/A118354/b118354.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A118354 Since g.f. G=G(x) of A118351 satisfies: G = 1 - 3*x*G + 3*x*G^2 + x*G^3 then
%F A118354 T(n,k) = T(n-1,k) - 3*T(n-1,k-1) + 3*T(n,k-1) + T(n+1,k-1).
%F A118354 Recurrence involving antidiagonals:
%F A118354 T(n,k) = T(n-1,k) + Sum_{j=1..k} [4*T(n-1+j,k-j) - 3*T(n-2+j,k-j)] for n>k>=0.
%e A118354 Show: T(n,k) = T(n-1,k) - 3*T(n-1,k-1) + 3*T(n,k-1) + T(n+1,k-1)
%e A118354 at n=8,k=4: T(8,4) = T(7,4) - 3*T(7,3) + 3*T(8,3) + T(9,3)
%e A118354 or: 2093 = 1353 - 3*244 + 3*340 + 452.
%e A118354 Triangle begins:
%e A118354   1;
%e A118354   1, 0;
%e A118354   1, 1,  0;
%e A118354   1, 2,  6,   0;
%e A118354   1, 3, 13,  42,    0;
%e A118354   1, 4, 21,  96,  325,     0;
%e A118354   1, 5, 30, 163,  770,  2688,      0;
%e A118354   1, 6, 40, 244, 1353,  6530,  23286,      0;
%e A118354   1, 7, 51, 340, 2093, 11760,  57612, 208659,       0;
%e A118354   1, 8, 63, 452, 3010, 18636, 105681, 523446, 1918314,        0;
%e A118354   1, 9, 76, 581, 4125, 27441, 170580, 973953, 4864795, 17994264, 0; ...
%t A118354 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 A118354 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 18 2021 *)
%o A118354 (PARI) {T(n,k)=polcoeff((serreverse(x*(1-3*x+sqrt((1-3*x)*(1-7*x)+x*O(x^k)))/2/(1-3*x))/x)^(n-k),k)}
%o A118354 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o A118354 (Sage)
%o A118354 @CachedFunction
%o A118354 def T(n, k):
%o A118354     if (k==0): return 1
%o A118354     elif (k==n): return 0
%o A118354     else: return T(n-1, k) - 3*T(n-1, k-1) + 3*T(n, k-1) + T(n+1, k-1)
%o A118354 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 18 2021
%Y A118354 Cf. A118350, A118351, A118352, A118353.
%Y A118354 Row sums: A151616.
%K A118354 nonn,tabl
%O A118354 0,8
%A A118354 _Paul D. Hanna_, Apr 26 2006