A139459
Triangle read by rows: binomial(3*n,3*k), 0 <= k <= n.
Original entry on oeis.org
1, 1, 1, 1, 20, 1, 1, 84, 84, 1, 1, 220, 924, 220, 1, 1, 455, 5005, 5005, 455, 1, 1, 816, 18564, 48620, 18564, 816, 1, 1, 1330, 54264, 293930, 293930, 54264, 1330, 1, 1, 2024, 134596, 1307504, 2704156, 1307504, 134596, 2024, 1, 1, 2925, 296010, 4686825, 17383860, 17383860, 4686825, 296010, 2925, 1
Offset: 0
First few rows of the triangle are:
[0] 1;
[1] 1, 1;
[2] 1, 20, 1;
[3] 1, 84, 84, 1;
[4] 1, 220, 924, 220, 1;
[5] 1, 455, 5005, 5005, 455, 1;
[6] 1, 816, 18564, 48620, 18564, 816, 1;
...
Row 5 = (1, 220, 924, 220, 1) = ConvOffs transform of (1, 20, 84, 220); where A006566 = (0, 1, 20, 84, 220, 455, ...).
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Table[Binomial[3*n, 3*k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 01 2025 *)
A177809
Symmetrical sequence:Binomial(n,5*m).
Original entry on oeis.org
1, 1, 1, 1, 252, 1, 1, 3003, 3003, 1, 1, 15504, 184756, 15504, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 142506, 30045015, 155117520, 30045015, 142506, 1, 1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1, 1, 658008, 847660528, 40225345056
Offset: 0
{1},
{1, 1},
{1, 252, 1},
{1, 3003, 3003, 1},
{1, 15504, 184756, 15504, 1},
{1, 53130, 3268760, 3268760, 53130, 1},
{1, 142506, 30045015, 155117520, 30045015, 142506, 1},
{1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1},
{1, 658008, 847660528, 40225345056, 137846528820, 40225345056, 847660528, 658008, 1},
{1, 1221759, 3190187286, 344867425584, 3169870830126, 3169870830126, 344867425584, 3190187286, 1221759, 1},
{1, 2118760, 10272278170, 2250829575120, 47129212243960, 126410606437752, 47129212243960, 2250829575120, 10272278170, 2118760, 1}
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t[n_, m_] = Binomial[n, 5*m];
Table[Table[t[n, m], {m, 0, Floor[n/5]}], {n, 0, 50, 5}];
Flatten[%]
A177810
Triangle binomial(6*n,6*m), 0 <= m <= n, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 924, 1, 1, 18564, 18564, 1, 1, 134596, 2704156, 134596, 1, 1, 593775, 86493225, 86493225, 593775, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 5245786, 11058116888, 353697121050, 353697121050, 11058116888, 5245786, 1
Offset: 0
1;
1, 1;
1, 924, 1;
1, 18564, 18564, 1;
1, 134596, 2704156, 134596, 1;
1, 593775, 86493225, 86493225, 593775, 1;
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t[n_, m_] := Binomial[n, 6*m]; Flatten@Table[Table[t[n, m], {m, 0, n/6}], {n, 0, 42, 6}]
Showing 1-3 of 3 results.
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