A391467 - OEIS

A391467

Triangle read by rows: T(n,0) = T(n,n) = 0 and T(n,k) = T(n, k-1) + T(n-1, k-1) + T(n-2, k-1) + k for 0 < k < n.

0, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 1, 5, 12, 0, 0, 1, 5, 17, 33, 0, 0, 1, 5, 18, 51, 89, 0, 0, 1, 5, 18, 57, 146, 241, 0, 0, 1, 5, 18, 58, 171, 412, 660, 0, 0, 1, 5, 18, 58, 178, 501, 1161, 1829, 0, 0, 1, 5, 18, 58, 179, 534, 1454, 3283, 5121, 0, 0, 1, 5, 18, 58, 179, 542, 1584, 4207, 9328, 14459, 0

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OFFSET

0,9

LINKS

Table of n, a(n) for n=0..77.

FORMULA

T(n+1, n) + (n+1) = A211278(n).

Sum_{k=0..n+1} T(2*n+2, k) = A052150(n).

Sum_{k=0..n} T(n+k, k) (mod 3) = A130481(n).

For n >= 2*k >= 0, T(n, k) = (3*(3^k-1)-2*k)/4.

EXAMPLE

Triangle T(n, k) starts:

n\k : 0 1 2 3 4 5 6 7 8 9

=================================================================

0 : 0

1 : 0 0

2 : 0 1 0

3 : 0 1 4 0

4 : 0 1 5 12 0

5 : 0 1 5 17 33 0

6 : 0 1 5 18 51 89 0

7 : 0 1 5 18 57 146 241 0

8 : 0 1 5 18 58 171 412 660 0

9 : 0 1 5 18 58 178 501 1161 1829 0

MAPLE

T := proc (n, k) option remember; if k = 0 or k = n then 0 else T(n, k-1)+T(n-1, k-1)+T(n-2, k-1) + k end if end proc:

seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);

MATHEMATICA

T[n_, k_]:=If[k==0||k==n, 0, T[n, k-1]+T[n-1, k-1]+T[n-2, k-1]+k]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* James C. McMahon, Dec 17 2025 *)

CROSSREFS

Cf. A000340 (central terms), A052150, A130481, A211278, A387263 (row sums), A390369.

Sequence in context: A254156 A344386 A046783 * A134832 A123163 A194794

Adjacent sequences: A391464 A391465 A391466 * A391468 A391469 A391470

KEYWORD

nonn,easy,tabl

AUTHOR

Mélika Tebni, Dec 10 2025

STATUS

approved