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%I #6 Jan 04 2026 20:28:11
%S 1,1,2,1,5,7,1,17,31,38,1,83,178,254,292,1,578,1330,2099,2683,2975,1,
%T 5474,12748,21338,29425,35375,38350,1,66932,153136,263870,382966,
%U 489383,566083,604433,1,1014002,2249503,3913556,5865952,7837136,9538360,10747226,11351659
%N Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + Sum_{j=0..k} binomial(k+1,j)*A(n-1,j)*A(k-j,0) with A(0,k) = 1.
%F Conjecture: A(n,0) = A233335(n+1).
%e Array begins:
%e ============================================================================
%e n\k| 0 1 2 3 4 5 6 ...
%e ---+------------------------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 ...
%e 1 | 2 5 17 83 578 5474 66932 ...
%e 2 | 7 31 178 1330 12748 153136 2249503 ...
%e 3 | 38 254 2099 21338 263870 3913556 68599898 ...
%e 4 | 292 2683 29425 382966 5865952 104739142 2158835605 ...
%e 5 | 2975 35375 489383 7837136 144383006 3039205823 72586944074 ...
%e 6 | 38350 566083 9538360 182707696 3959710333 96600644167 2639032949860 ...
%e ...
%o (PARI) antidiagonals(n) = {my(v = vector(n+1, i, vector(n-i+2, j, i==1)));
%o for(i=1, n, forstep(j=i-1, 0, -1, v[i-j+1][j+1] = v[i-j][j+2] + sum(k=0, j, binomial(j+1,k)*v[i-j][k+1]*v[j-k+1][1])));
%o v = vector(n+1, i, vector(i, j, v[j][i-j+1]))}
%Y Cf. A233335.
%K nonn,tabl
%O 0,3
%A _Mikhail Kurkov_, Dec 29 2025