A374244 A Catalan-like sequence formed from the row sums of a Catalan-like triangle where row n is truncated to have ceiling((n+4)*log(3)/log(2)) - (n + 6) terms.

1, 2, 5, 9, 23, 43, 113, 331, 698, 1966, 4072, 11433, 23701, 66734, 205712, 459632, 1348864, 2927822, 8499580, 26809375, 61495590, 183946295, 408179706, 1204202538, 2643267587

Offset

1,2

Links

Table of n, a(n) for n=1..25.

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) where j is limited to the truncated length.

Example

Standard Catalan:
 n   Sum   Triangle terms
 1     1 = 1;
 2     2 = 1, 1;
 3     5 = 1, 2,  2;
 4    14 = 1, 3,  5; /5
 5    42 = 1, 4,  9, 14; /14
 6   132 = 1, 5, 14, 28; /42;  14
 7   429 = 1, 6, 20, 48, 90; /132; 132
...
When n=4, number of terms is restricted to 3 dropping 1 term; ceiling((4+4)*log(3)/log(2)) - (4 + 6) = 3.
When n=6, number of terms is restricted to 4 dropping 2 terms; ceiling((6+4)*log(3)/log(2)) - (6 + 6) = 4.
etc.
Truncating at the point indicated by / and summing the remaining triangle terms in the normal way results in:
n    Sum  Truncated Triangle terms
 1     1 = 1;
 2     2 = 1, 1;
 3     5 = 1, 2, 2;
 4     9 = 1, 3, 5;
 5    23 = 1, 4, 9, 9;
 6    43 = 1, 5, 14, 23;
 7   113 = 1, 6, 20, 43, 43;
 8   331 = 1, 7, 27, 70, 113, 113;
 9   698 = 1, 8, 35, 105, 218, 331;
10  1966 = 1, 9, 44, 149, 367, 698, 698;
11  4072 = 1, 10, 54, 203, 570, 1268, 1966;
12 11433 = 1, 11, 65, 268, 838, 2106, 4072, 4072;
13 23701 = 1, 12, 77, 345, 1183, 3289, 7361, 11433;
...

Prog

(PARI) f(n) = {
my(terms(j)=ceil((j+4)*log(3)/log(2)) - (j+6));
my(T=vector(n));
my(S=vector(n));
for(y=1, n,
       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]);
    );
for(y=1, n, print(S[y], ": ", T[y]); );
}

Crossrefs

Cf. A009766, A000108, Half Catalan A000992.

Sequence in context: A342556 A002935 A128266 * A088356 A246350 A192477

Adjacent sequences: A374241 A374242 A374243 * A374245 A374246 A374247

Keyword

nonn,easy

Author

Rob Bunce, Jul 01 2024

Status

approved