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

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

Examples

			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, -, -, -;
...
		

Crossrefs

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

Programs

PARI

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;
}

Formula

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.