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A382851 a(n) = least number in row n of Pascal's triangle that exceeds every number in row n-1.

%I A382851 #12 May 02 2025 23:59:41
%S A382851 2,3,4,10,15,21,56,84,210,330,495,1287,2002,5005,8008,19448,31824,
%T A382851 50388,125970,203490,497420,817190,1961256,3268760,5311735,13037895,
%U A382851 21474180,51895935,86493225,206253075,347373600,818809200,1391975640,3247943160,5567902560
%N A382851 a(n) = least number in row n of Pascal's triangle that exceeds every number in row n-1.
%e A382851 Rows 0 to 5 of Pascal's triangle:
%e A382851   1
%e A382851   1  1
%e A382851   1  2   1
%e A382851   1  3   3   1
%e A382851   1  4   6   4  1
%e A382851   1  5  10  10  5  1,
%e A382851 10 is the least number in row 5 that exceeds max{1,4,6}, so a(5)=10
%t A382851 z = 40; c[n_, k_] := Binomial[n, k];
%t A382851 t[n_] := Table[c[n, k], {k, 0, n}];
%t A382851 a[n_] := Select[Range[z], c[n, #] > c[n - 1, Floor[(n - 1)/2]] &, 1];
%t A382851 Flatten[Table[a[n], {n, 1, 3 z}]]  (* A382850 *)
%t A382851 Flatten[Table[c[n, a[n]], {n, 1, z}]]  (* A382851 *)
%t A382851 a[n_] := Block[{b, k = 1, m = Binomial[n -1, Floor[(n -1)/2]]}, While[b = Binomial[n, k]; b < m, k++]; b]; Array[a, 35, 2] (* _Robert G. Wilson v_, May 02 2025 *)
%o A382851 (PARI) row(n) = vector(n+1, k, binomial(n,k-1));
%o A382851 a(n) = my(val = vecmax(row(n-1)), w = row(n)); for (i=1, #w, if (w[i] > val, return(w[i]));); \\ _Michel Marcus_, Apr 13 2025
%Y A382851 Cf. A007318, A382850.
%K A382851 nonn
%O A382851 2,1
%A A382851 _Clark Kimberling_, Apr 13 2025