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).

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

Author

Rob Bunce, Sep 20 2024

Keywords

• nonn
• easy

Comments

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.

Examples

			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, -, -, -;
...
		

Crossrefs

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

Programs

PARI

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;
}

Formula

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.