A374244 -id:A374244 - cp's OEIS frontend

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).

1, 4, 14, 34, 103, 228, 665, 2096, 4787, 14239, 31330, 91728, 199328, 580128, 1834665, 4223092, 12667903, 28207395, 83435822, 267154051, 623837740, 1891453021, 4265101202, 12735718304, 28359351604, 84126071303, 270338873771, 634653510356, 1933488496208

Offset: 1

Rob Bunce, Sep 20 2024

a(1) = 1, all other rows are summed following application of the truncation formula.

Equivalent to truncation of A002057 starting from the n(4) term.

			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.
When n=9, number of terms is restricted to 8, dropping 3 terms; ceiling((9+4)*log(3)/log(2)) - (9+4) = 8.
etc.
Truncating A002057 at this point, with dropped terms indicated by - and summing the remaining triangle terms in the normal way results in:
 n   sum   truncated triangle terms
 1     1 = 1;
 2     4 = 1, 1,  1,   1;
 3    14 = 1, 2,  3,   4,   4;
 4    34 = 1, 3,  6,  10,  14,    -;
 5   103 = 1, 4, 10,  20,  34,   34,    -;
 6   228 = 1, 5, 15,  35,  69,  103,    -,    -;
 7   665 = 1, 6, 21,  56, 125,  228,  228,    -,    -;
 8  2096 = 1, 7, 28,  84, 209,  437,  665,  665,    -, -;
 9  4787 = 1, 8, 36, 120, 329,  766, 1431, 2096,    -, -, -;
10 14239 = 1, 9, 45, 165, 494, 1260, 2691, 4787, 4787, -, -, -;
...

Cf. A009766, A000108, A002057, A374244, Half Catalan A000992.

lista(nn) = {
my(terms(j)=ceil((j+4)*log(3)/log(2)) - (j+4));
my(T=vector(nn));
my(S=vector(nn));
for(y=1, nn,
  if(y==1,
      T[1]=[1];
      S[1]=1		
    ,
      my(k=terms(y));
      T[y]=vector(k);
      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])));
      S[y]=vecsum(T[y])
    );
  );
S;
}

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.

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).

Original entry on oeis.org

1, 2, 5, 14, 28, 76, 151, 412, 1239, 2689, 7724, 16351, 46607, 98276, 280035, 871218, 1967577, 5819850, 12749014, 37260985, 118163637, 272787542, 819934670, 1829959304, 5422130623, 11963162678, 35243160809, 112614062317, 262572197079, 795710438547, 1794155974237

Offset: 1

Rob Bunce, Sep 20 2024

a(1) = 1, all other rows are summed following application of the truncation formula.

Equivalent to summing the remaining terms after truncation of A009766 starting from the 5th row.

			When n=6, number of terms is restricted to 5 dropping 1 term; ceiling((6+3)*log(3)/log(2)) - (6+4) = 5.
When n=10, number of terms is restricted to 7 dropping 3 terms; ceiling((10+3)*log(3)/log(2)) - (10+4) = 7.
Truncating A009766 at the point indicated by - and summing the remaining triangle terms in the normal way results in:
row   sum   truncated triangle terms
 1     1 = 1;
 2     2 = 1, 1;
 3     5 = 1, 2, 2;
 4    14 = 1, 3, 5, 5;
 5    28 = 1, 4, 9, 14, -;
 6    76 = 1, 5, 14, 28, 28, -;
 7   151 = 1, 6, 20, 48, 76, -, -;
 8   412 = 1, 7, 27, 75, 151, 151, -, -;
 9  1239 = 1, 8, 35, 110, 261, 412, 412, -, -;
10  2689 = 1, 9, 44, 154, 415, 827, 1239, -, -, -;
...

Cf. A000108, A009766, A000108, A374244, A000992 (half Catalan).

lista(nn) = {
my(terms(j)=ceil((j+3)*log(3)/log(2)) - (j+4));
my(T=vector(nn));
my(S=vector(nn));
for(y=1, nn,
  if(y==1,
      T[1]=[1];
      S[1]=1		
    ,
      my(k=terms(y));
      T[y]=vector(k);
      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])));
      S[y]=vecsum(T[y])
    );
  );
  S;
}

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.

cp's OEIS Frontend

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).

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