OFFSET
1,2
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
LINKS
Sean A. Irvine, Java program (github)
FORMULA
1 plus T(n,k) for n >= 2 and 2 <= k <= n, duplicates removed and sorted, where T(n,k) is the triangle of trinomial coefficients (A027907). - Chai Wah Wu, Apr 27 2026
a(n) = n^2/2 - b*n^(5/3) - c*n^(3/2) + O(n^(7/5)), where b = 3^(1/3) and c = 12^(1/4). - Charles R Greathouse IV, Apr 28 2026
PROG
(Python)
from itertools import count, islice
from math import comb
from heapq import heappop, heappush
def A035000_gen(): # generator of terms
h, hset = [1], {1}
for n in count(2):
yield heappop(h)
for k in range(2, n+1):
t = sum(comb(n, m:=2*i+n-k)*comb(m, i) for i in range((k>>1)+1))
if t not in hset:
heappush(h, t)
hset.add(t)
(PARI) T(n, k) = sum(i = 0, k\2, binomial(n, 2*i + n - k) * binomial(2*i + n - k, i))
list(lim)=lim\=1; my(v=List([1])); for(n=2, (sqrtint(8*lim+1)-1)\2, for(k=2, n, my(t=T(n, k)); if(t>lim, break); listput(v, t))); Set(v) \\ Charles R Greathouse IV, Apr 28 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved
