A064282 Triangle read by rows: T(n,k) = binomial(3n+3, k)*(n-k+1)/(n+1).

1, 1, 3, 1, 6, 12, 1, 9, 33, 55, 1, 12, 63, 182, 273, 1, 15, 102, 408, 1020, 1428, 1, 18, 150, 760, 2565, 5814, 7752, 1, 21, 207, 1265, 5313, 15939, 33649, 43263, 1, 24, 273, 1950, 9750, 35880, 98670, 197340, 246675, 1, 27, 348, 2842, 16443, 71253, 237510

Offset: 0

Author

Henry Bottomley, Sep 24 2001

Keywords

• nonn
• tabl

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000

Crossrefs

Columns include A000012 and A008585. Right hand columns include A001764 and A006630. Row sums are A047099. Triangles A010054 (Triangle Indicator), A007318 (Pascal) and A050166 form a sequence which has this as its next member.

Programs

Mathematica

Flatten[Table[Binomial[3n+3,k] (n-k+1)/(n+1),{n,0,10},{k,0,n}]] (* Harvey P. Dale, Dec 26 2014 *)

PARI

{ n=-1; for (m=0, 10^9, for (k=0, m, a=binomial(3*m + 3, k)*(m - k + 1)/(m + 1); write("b064282.txt", n++, " ", a); if (n==1000, break)); if (n==1000, break) ) } \\ Harry J. Smith, Sep 11 2009

Formula

T(n, k) = T(n-1, k) + 3*T(n-1, k-1) + 3*T(n-1, k-2) + T(n-1, k-3) [starting with T(0,0)=1 and T(n,k)=0 if n < 0 or n < k].