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A374244 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.

%I A374244 #27 Nov 16 2024 17:36:57
%S A374244 1,2,5,9,23,43,113,331,698,1966,4072,11433,23701,66734,205712,459632,
%T A374244 1348864,2927822,8499580,26809375,61495590,183946295,408179706,
%U A374244 1204202538,2643267587,7756962475,24708004563,57390010121,173405214133,389606249120,1160606285961,3738436950162
%N A374244 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.
%H A374244 Andrew Howroyd, <a href="/A374244/b374244.txt">Table of n, a(n) for n = 1..1000</a>
%F A374244 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 A374244 Standard Catalan:
%e A374244  n   Sum   Triangle terms
%e A374244  1     1 = 1;
%e A374244  2     2 = 1, 1;
%e A374244  3     5 = 1, 2,  2;
%e A374244  4    14 = 1, 3,  5; /5
%e A374244  5    42 = 1, 4,  9, 14; /14
%e A374244  6   132 = 1, 5, 14, 28; /42;  14
%e A374244  7   429 = 1, 6, 20, 48, 90; /132; 132
%e A374244 ...
%e A374244 When n=4, number of terms is restricted to 3 dropping 1 term; ceiling((4+4)*log(3)/log(2)) - (4 + 6) = 3.
%e A374244 When n=6, number of terms is restricted to 4 dropping 2 terms; ceiling((6+4)*log(3)/log(2)) - (6 + 6) = 4.
%e A374244 etc.
%e A374244 Truncating at the point indicated by / and summing the remaining triangle terms in the normal way results in:
%e A374244 n    Sum  Truncated Triangle terms
%e A374244  1     1 = 1;
%e A374244  2     2 = 1, 1;
%e A374244  3     5 = 1, 2, 2;
%e A374244  4     9 = 1, 3, 5;
%e A374244  5    23 = 1, 4, 9, 9;
%e A374244  6    43 = 1, 5, 14, 23;
%e A374244  7   113 = 1, 6, 20, 43, 43;
%e A374244  8   331 = 1, 7, 27, 70, 113, 113;
%e A374244  9   698 = 1, 8, 35, 105, 218, 331;
%e A374244 10  1966 = 1, 9, 44, 149, 367, 698, 698;
%e A374244 11  4072 = 1, 10, 54, 203, 570, 1268, 1966;
%e A374244 12 11433 = 1, 11, 65, 268, 838, 2106, 4072, 4072;
%e A374244 13 23701 = 1, 12, 77, 345, 1183, 3289, 7361, 11433;
%e A374244 ...
%o A374244 (PARI) lista(N) = {
%o A374244   my(T=vector(N, n, vector(logint(3^(n+4), 2)-n-5)));
%o A374244   for(n=1, #T
%o A374244   , for(k=1, #T[n]
%o A374244     , T[n][k]= if(1==k, 1, k<=#T[n-1], T[n][k-1]+T[n-1][k], T[n][k-1])
%o A374244     );
%o A374244   );
%o A374244   vector(#T, n, vecsum(T[n]));
%o A374244 }
%Y A374244 Cf. A009766, A000108, Half Catalan A000992.
%Y A374244 Cf. A100982 (row sums of A368514).
%K A374244 nonn,easy
%O A374244 1,2
%A A374244 _Rob Bunce_, Jul 01 2024
%E A374244 a(26) onwards from _Andrew Howroyd_, Oct 25 2024