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

1,2

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.

Links

Table of n, a(n) for n=1..31.

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.

Example

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, -, -, -;
...

Prog

(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;
}

Crossrefs

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

Sequence in context: A212340 A304719 A022630 * A047133 A031874 A120328

Adjacent sequences: A376322 A376323 A376324 * A376326 A376327 A376328

Keyword

nonn,easy

Author

Rob Bunce, Sep 20 2024

Status

approved