%I #15 Aug 03 2024 11:57:05
%S 1,2,5,9,23,43,113,331,698,1966,4072,11433,23701,66734,205712,459632,
%T 1348864,2927822,8499580,26809375,61495590,183946295,408179706,
%U 1204202538,2643267587
%N A Catalan-like sequence formed from the row sums of a Catalan-like triangle where row n is truncated to have ceiling((n+4)*log(3)/log(2)) - (n + 6) terms.
%F Same as for a normal Catalan triangle T(n,k), read by rows, each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j) where j is limited to the truncated length.
%e Standard Catalan:
%e n Sum Triangle terms
%e 1 1 = 1;
%e 2 2 = 1, 1;
%e 3 5 = 1, 2, 2;
%e 4 14 = 1, 3, 5; /5
%e 5 42 = 1, 4, 9, 14; /14
%e 6 132 = 1, 5, 14, 28; /42; 14
%e 7 429 = 1, 6, 20, 48, 90; /132; 132
%e ...
%e When n=4, number of terms is restricted to 3 dropping 1 term; ceiling((4+4)*log(3)/log(2)) - (4 + 6) = 3.
%e When n=6, number of terms is restricted to 4 dropping 2 terms; ceiling((6+4)*log(3)/log(2)) - (6 + 6) = 4.
%e etc.
%e Truncating at the point indicated by / and summing the remaining triangle terms in the normal way results in:
%e n Sum Truncated Triangle terms
%e 1 1 = 1;
%e 2 2 = 1, 1;
%e 3 5 = 1, 2, 2;
%e 4 9 = 1, 3, 5;
%e 5 23 = 1, 4, 9, 9;
%e 6 43 = 1, 5, 14, 23;
%e 7 113 = 1, 6, 20, 43, 43;
%e 8 331 = 1, 7, 27, 70, 113, 113;
%e 9 698 = 1, 8, 35, 105, 218, 331;
%e 10 1966 = 1, 9, 44, 149, 367, 698, 698;
%e 11 4072 = 1, 10, 54, 203, 570, 1268, 1966;
%e 12 11433 = 1, 11, 65, 268, 838, 2106, 4072, 4072;
%e 13 23701 = 1, 12, 77, 345, 1183, 3289, 7361, 11433;
%e ...
%o (PARI) f(n) = {
%o my(terms(j)=ceil((j+4)*log(3)/log(2)) - (j+6));
%o my(T=vector(n));
%o my(S=vector(n));
%o for(y=1, n,
%o my(k=terms(y));
%o T[y]=vector(k);
%o for(i=1, k, if(i==1,T[y][i]=1,if(i<=length(T[y-1]),T[y][i]=T[y-1][i]+T[y][i-1],T[y][i]=T[y][i-1])));
%o S[y]=vecsum(T[y]);
%o );
%o for(y=1, n, print(S[y], ": ", T[y]); );
%o }
%Y Cf. A009766, A000108, Half Catalan A000992.
%K nonn,easy
%O 1,2
%A _Rob Bunce_, Jul 01 2024