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, 7756962475, 24708004563, 57390010121, 173405214133, 389606249120, 1160606285961, 3738436950162
Offset: 1
Examples
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; ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1000
Programs
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PARI
lista(N) = { my(T=vector(N, n, vector(logint(3^(n+4), 2)-n-5))); for(n=1, #T , for(k=1, #T[n] , T[n][k]= if(1==k, 1, k<=#T[n-1], T[n][k-1]+T[n-1][k], T[n][k-1]) ); ); vector(#T, n, vecsum(T[n])); }
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.
Extensions
a(26) onwards from Andrew Howroyd, Oct 25 2024