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%I #4 Jan 04 2026 21:07:20
%S 1,1,2,1,4,6,1,10,18,24,1,34,64,94,118,1,152,278,414,556,674,1,826,
%T 1438,2096,2842,3634,4308,1,5134,8528,12046,16110,20772,25754,30062,1,
%U 35196,56202,76978,100788,128750,160834,195204,225266,1,260462,402638,536728,687578,864906,1073180,1311370,1566698,1791964
%N Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + Sum_{j=0..k} A(n-1,j)*A(k-j,0) with A(0,k) = 1.
%F Conjecture: A(n,0) = A088713(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 4 10 34 152 826 5134 ...
%e 2 | 6 18 64 278 1438 8528 56202 ...
%e 3 | 24 94 414 2096 12046 76978 536728 ...
%e 4 | 118 556 2842 16110 100788 687578 5052062 ...
%e 5 | 674 3634 20772 128750 864906 6251224 48200006 ...
%e 6 | 4308 25754 160834 1073180 7665010 58335158 470209888 ...
%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, 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. A088713.
%K nonn,tabl
%O 0,3
%A _Mikhail Kurkov_, Dec 30 2025