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A118343 Triangle, read by rows, where diagonals are successive self-convolutions of A108447.

%I A118343 #8 Mar 18 2021 11:28:46
%S A118343 1,1,0,1,1,0,1,2,4,0,1,3,9,20,0,1,4,15,48,113,0,1,5,22,85,282,688,0,1,
%T A118343 6,30,132,519,1762,4404,0,1,7,39,190,837,3330,11488,29219,0,1,8,49,
%U A118343 260,1250,5516,22135,77270,199140,0,1,9,60,343,1773,8461,37404,151089,532239,1385904,0
%N A118343 Triangle, read by rows, where diagonals are successive self-convolutions of A108447.
%C A118343 A108447 equals the central terms of pendular triangle A118340 and the diagonals of this triangle form the semi-diagonals of the triangle A118340. Row sums equal A054727, the number of forests of rooted trees with n nodes on a circle without crossing edges.
%H A118343 G. C. Greubel, <a href="/A118343/b118343.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A118343 Since g.f. G=G(x) of A108447 satisfies: G = 1 - x*G + x*G^2 + x*G^3 then T(n,k) = T(n-1,k) - T(n-1,k-1) + T(n,k-1) + T(n+1,k-1). Also, a recurrence involving antidiagonals is: T(n,k) = T(n-1,k) + Sum_{j=1..k} [2*T(n-1+j,k-j) - T(n-2+j,k-j)] for n>k>=0.
%F A118343 Sum_{k=0..n} T(n,k) = [n=0] + A054727(n) = [n=0] + Sum_{j=1..n} binomial(n, j-1)*binomial(3*n-2*j-1, n-j)/(2*n-j). - _G. C. Greubel_, Mar 17 2021
%e A118343 Show: T(n,k) = T(n-1,k) - T(n-1,k-1) + T(n,k-1) + T(n+1,k-1)
%e A118343 at n=8,k=4: T(8,4) = T(7,4) - T(7,3) + T(8,3) + T(9,3)
%e A118343 or 837 = 519 - 132 + 190 + 260.
%e A118343 Triangle begins:
%e A118343   1;
%e A118343   1, 0;
%e A118343   1, 1,  0;
%e A118343   1, 2,  4,   0;
%e A118343   1, 3,  9,  20,    0;
%e A118343   1, 4, 15,  48,  113,    0;
%e A118343   1, 5, 22,  85,  282,  688,     0;
%e A118343   1, 6, 30, 132,  519, 1762,  4404,      0;
%e A118343   1, 7, 39, 190,  837, 3330, 11488,  29219,      0;
%e A118343   1, 8, 49, 260, 1250, 5516, 22135,  77270, 199140,       0;
%e A118343   1, 9, 60, 343, 1773, 8461, 37404, 151089, 532239, 1385904, 0;
%p A118343 T:= proc(n, k) option remember;
%p A118343       if k<0 or  k>n then 0;
%p A118343     elif k=0 then 1;
%p A118343     elif k=n then 0;
%p A118343     else T(n-1, k) -T(n-1, k-1) +T(n, k-1) +T(n+1, k-1);
%p A118343       fi; end:
%p A118343 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 17 2021
%t A118343 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, 1, If[k==n, 0, T[n-1, k] -T[n-1, k-1] +T[n, k-1] +T[n+1, k-1] ]]];
%t A118343 Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Mar 17 2021 *)
%o A118343 (PARI) {T(n,k)=polcoeff((serreverse(x*(1-x+sqrt((1-x)*(1-5*x)+x*O(x^k)))/2/(1-x))/x)^(n-k),k)}
%o A118343 (Sage)
%o A118343 @CachedFunction
%o A118343 def T(n, k):
%o A118343     if (k<0 or k>n): return 0
%o A118343     elif (k==0): return 1
%o A118343     elif (k==n): return 0
%o A118343     else: return T(n-1, k) -T(n-1, k-1) +T(n, k-1) +T(n+1, k-1)
%o A118343 flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 17 2021
%Y A118343 Cf. A054727 (row sums), A108447, A118340.
%K A118343 nonn,tabl
%O A118343 0,8
%A A118343 _Paul D. Hanna_, Apr 26 2006