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A376319 A Catalan-like sequence formed by summing the truncation of the terms of the fourth convolution of the Catalan Triangle where the number of row terms are truncated to ceiling((n+4)*log(3)/log(2)) - (n+4).

%I A376319 #17 Oct 23 2024 01:21:59
%S A376319 1,4,14,34,103,228,665,2096,4787,14239,31330,91728,199328,580128,
%T A376319 1834665,4223092,12667903,28207395,83435822,267154051,623837740,
%U A376319 1891453021,4265101202,12735718304,28359351604,84126071303,270338873771,634653510356,1933488496208
%N A376319 A Catalan-like sequence formed by summing the truncation of the terms of the fourth convolution of the Catalan Triangle where the number of row terms are truncated to ceiling((n+4)*log(3)/log(2)) - (n+4).
%C A376319 a(1) = 1, all other rows are summed following application of the truncation formula.
%C A376319 Equivalent to truncation of A002057 starting from the n(4) term.
%F A376319 Same as for a normal fourth convolution 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 A376319 When n=6, number of terms is restricted to 6, dropping 2 terms from the standard triangle; ceiling((6+4)*log(3)/log(2)) - (6+4) = 6.
%e A376319 When n=9, number of terms is restricted to 8, dropping 3 terms; ceiling((9+4)*log(3)/log(2)) - (9+4) = 8.
%e A376319 etc.
%e A376319 Truncating A002057 at this point, with dropped terms indicated by - and summing the remaining triangle terms in the normal way results in:
%e A376319  n   sum   truncated triangle terms
%e A376319  1     1 = 1;
%e A376319  2     4 = 1, 1,  1,   1;
%e A376319  3    14 = 1, 2,  3,   4,   4;
%e A376319  4    34 = 1, 3,  6,  10,  14,    -;
%e A376319  5   103 = 1, 4, 10,  20,  34,   34,    -;
%e A376319  6   228 = 1, 5, 15,  35,  69,  103,    -,    -;
%e A376319  7   665 = 1, 6, 21,  56, 125,  228,  228,    -,    -;
%e A376319  8  2096 = 1, 7, 28,  84, 209,  437,  665,  665,    -, -;
%e A376319  9  4787 = 1, 8, 36, 120, 329,  766, 1431, 2096,    -, -, -;
%e A376319 10 14239 = 1, 9, 45, 165, 494, 1260, 2691, 4787, 4787, -, -, -;
%e A376319 ...
%o A376319 (PARI) lista(nn) = {
%o A376319 my(terms(j)=ceil((j+4)*log(3)/log(2)) - (j+4));
%o A376319 my(T=vector(nn));
%o A376319 my(S=vector(nn));
%o A376319 for(y=1, nn,
%o A376319   if(y==1,
%o A376319       T[1]=[1];
%o A376319       S[1]=1		
%o A376319     ,
%o A376319       my(k=terms(y));
%o A376319       T[y]=vector(k);
%o A376319       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 A376319       S[y]=vecsum(T[y])
%o A376319     );
%o A376319   );
%o A376319 S;
%o A376319 }
%Y A376319 Cf. A009766, A000108, A002057, A374244, Half Catalan A000992.
%K A376319 nonn,easy
%O A376319 1,2
%A A376319 _Rob Bunce_, Sep 20 2024