This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376325 #15 Oct 23 2024 01:22:58
%S A376325 1,2,5,14,28,76,151,412,1239,2689,7724,16351,46607,98276,280035,
%T A376325 871218,1967577,5819850,12749014,37260985,118163637,272787542,
%U A376325 819934670,1829959304,5422130623,11963162678,35243160809,112614062317,262572197079,795710438547,1794155974237
%N A376325 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 A376325 a(1) = 1, all other rows are summed following application of the truncation formula.
%C A376325 Equivalent to summing the remaining terms after truncation of A009766 starting from the 5th row.
%F A376325 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 A376325 When n=6, number of terms is restricted to 5 dropping 1 term; ceiling((6+3)*log(3)/log(2)) - (6+4) = 5.
%e A376325 When n=10, number of terms is restricted to 7 dropping 3 terms; ceiling((10+3)*log(3)/log(2)) - (10+4) = 7.
%e A376325 Truncating A009766 at the point indicated by - and summing the remaining triangle terms in the normal way results in:
%e A376325 row sum truncated triangle terms
%e A376325 1 1 = 1;
%e A376325 2 2 = 1, 1;
%e A376325 3 5 = 1, 2, 2;
%e A376325 4 14 = 1, 3, 5, 5;
%e A376325 5 28 = 1, 4, 9, 14, -;
%e A376325 6 76 = 1, 5, 14, 28, 28, -;
%e A376325 7 151 = 1, 6, 20, 48, 76, -, -;
%e A376325 8 412 = 1, 7, 27, 75, 151, 151, -, -;
%e A376325 9 1239 = 1, 8, 35, 110, 261, 412, 412, -, -;
%e A376325 10 2689 = 1, 9, 44, 154, 415, 827, 1239, -, -, -;
%e A376325 ...
%o A376325 (PARI) lista(nn) = {
%o A376325 my(terms(j)=ceil((j+3)*log(3)/log(2)) - (j+4));
%o A376325 my(T=vector(nn));
%o A376325 my(S=vector(nn));
%o A376325 for(y=1, nn,
%o A376325 if(y==1,
%o A376325 T[1]=[1];
%o A376325 S[1]=1
%o A376325 ,
%o A376325 my(k=terms(y));
%o A376325 T[y]=vector(k);
%o A376325 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 A376325 S[y]=vecsum(T[y])
%o A376325 );
%o A376325 );
%o A376325 S;
%o A376325 }
%Y A376325 Cf. A000108, A009766, A000108, A374244, A000992 (half Catalan).
%K A376325 nonn,easy
%O A376325 1,2
%A A376325 _Rob Bunce_, Sep 20 2024