%I #15 Oct 23 2024 01:22:58
%S 1,2,5,14,28,76,151,412,1239,2689,7724,16351,46607,98276,280035,
%T 871218,1967577,5819850,12749014,37260985,118163637,272787542,
%U 819934670,1829959304,5422130623,11963162678,35243160809,112614062317,262572197079,795710438547,1794155974237
%N A Catalan-like sequence formed by summing the truncation of the terms of a Catalan Triangle A009766 where the number of row terms are truncated to ceiling((n+3)*log(3)/log(2)) - (n+4).
%C a(1) = 1, all other rows are summed following application of the truncation formula.
%C Equivalent to summing the remaining terms after truncation of A009766 starting from the 5th row.
%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) but where j is limited to the truncated length.
%e When n=6, number of terms is restricted to 5 dropping 1 term; ceiling((6+3)*log(3)/log(2)) - (6+4) = 5.
%e When n=10, number of terms is restricted to 7 dropping 3 terms; ceiling((10+3)*log(3)/log(2)) - (10+4) = 7.
%e Truncating A009766 at the point indicated by - and summing the remaining triangle terms in the normal way results in:
%e row sum truncated 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 28 = 1, 4, 9, 14, -;
%e 6 76 = 1, 5, 14, 28, 28, -;
%e 7 151 = 1, 6, 20, 48, 76, -, -;
%e 8 412 = 1, 7, 27, 75, 151, 151, -, -;
%e 9 1239 = 1, 8, 35, 110, 261, 412, 412, -, -;
%e 10 2689 = 1, 9, 44, 154, 415, 827, 1239, -, -, -;
%e ...
%o (PARI) lista(nn) = {
%o my(terms(j)=ceil((j+3)*log(3)/log(2)) - (j+4));
%o my(T=vector(nn));
%o my(S=vector(nn));
%o for(y=1, nn,
%o if(y==1,
%o T[1]=[1];
%o S[1]=1
%o ,
%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 );
%o S;
%o }
%Y Cf. A000108, A009766, A000108, A374244, A000992 (half Catalan).
%K nonn,easy,new
%O 1,2
%A _Rob Bunce_, Sep 20 2024