A360560
Triangle read by rows. T(n, k) = (1/2) * C(n, k) * C(3*n - 1, n) for n > 0 and T(0, 0) = 1.
Original entry on oeis.org
1, 1, 1, 5, 10, 5, 28, 84, 84, 28, 165, 660, 990, 660, 165, 1001, 5005, 10010, 10010, 5005, 1001, 6188, 37128, 92820, 123760, 92820, 37128, 6188, 38760, 271320, 813960, 1356600, 1356600, 813960, 271320, 38760, 245157, 1961256, 6864396, 13728792, 17160990, 13728792, 6864396, 1961256, 245157
Offset: 0
Triangle begins:
1;
1, 1;
5, 10, 5;
28, 84, 84, 28;
165, 660, 990, 660, 165;
1001, 5005, 10010, 10010, 5005, 1001;
-
T := (n, k) -> ifelse(n = 0, 1, binomial(n, k)*binomial(3*n - 1, n)/2):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
-
T(n,m):=1/2*binomial(n+1,m)*binomial(3*n+2,n+1);
A374440
Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with boundary conditions: if k = 0 or k = 2 then T = 1; if k = 1 then T = n - 1.
Original entry on oeis.org
1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 1, 1, 1, 1, 4, 1, 3, 2, 0, 1, 5, 1, 6, 3, 1, 1, 1, 6, 1, 10, 4, 4, 3, 0, 1, 7, 1, 15, 5, 10, 6, 1, 1, 1, 8, 1, 21, 6, 20, 10, 5, 4, 0, 1, 9, 1, 28, 7, 35, 15, 15, 10, 1, 1, 1, 10, 1, 36, 8, 56, 21, 35, 20, 6, 5, 0
Offset: 0
Triangle starts:
[ 0] 1;
[ 1] 1, 0;
[ 2] 1, 1, 1;
[ 3] 1, 2, 1, 0;
[ 4] 1, 3, 1, 1, 1;
[ 5] 1, 4, 1, 3, 2, 0;
[ 6] 1, 5, 1, 6, 3, 1, 1;
[ 7] 1, 6, 1, 10, 4, 4, 3, 0;
[ 8] 1, 7, 1, 15, 5, 10, 6, 1, 1;
[ 9] 1, 8, 1, 21, 6, 20, 10, 5, 4, 0;
[10] 1, 9, 1, 28, 7, 35, 15, 15, 10, 1, 1;
Cf.
A000032 (Lucas),
A001611 (even sums, Fibonacci + 1),
A000071 (odd sums, Fibonacci - 1),
A001911 (alternating sums, Fibonacci(n+3) - 2),
A025560 (row lcm),
A073028 (row max),
A117671 &
A025174 (central terms),
A057979 (subdiagonal),
A000217 (column 3).
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T := proc(n, k) option remember; if k = 0 or k = 2 then 1 elif k > n then 0
elif k = 1 then n - 1 else T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..9);
T := (n, k) -> ifelse(k = 0, 1, binomial(n - floor(k/2), ceil(k/2)) -
binomial(n - ceil((k + irem(k + 1, 2))/2), floor(k/2))):
A139796
Last term of A139687(n) with a fourth leading 1 = 1, 1, 1, 1, 2, 2, 1, 3, 5, 5 rows.
Original entry on oeis.org
1, 1, 2, 5, 9, 28, 48, 165, 275, 1001, 1638, 6188
Offset: 0
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