A029651 -id:A029651 - cp's OEIS frontend

A238160 A skewed version of triangular array A029653.

1, 0, 2, 0, 1, 2, 0, 0, 3, 2, 0, 0, 1, 5, 2, 0, 0, 0, 4, 7, 2, 0, 0, 0, 1, 9, 9, 2, 0, 0, 0, 0, 5, 16, 11, 2, 0, 0, 0, 0, 1, 14, 25, 13, 2, 0, 0, 0, 0, 0, 6, 30, 36, 15, 2, 0, 0, 0, 0, 0, 1, 20, 55, 49, 17, 2, 0, 0, 0, 0, 0, 0, 7, 50, 91, 64, 19, 2, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Feb 18 2014

Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are Fib(n+2).
Column sums are A003945(k).
Diagonal sums are (-1)^(n+1)*A109266(n+1).
T(3*n,2*n) = A029651(n).

			Triangle begins:
1;
0, 2;
0, 1, 2;
0, 0, 3, 2;
0, 0, 1, 5, 2;
0, 0, 0, 4, 7, 2;
0, 0, 0, 1, 9, 9, 2;
0, 0, 0, 0, 5, 16, 11, 2;
0, 0, 0, 0, 1, 14, 25, 13, 2;
0, 0, 0, 0, 0, 6, 30, 36, 15, 2;
0, 0, 0, 0, 0, 1, 20, 55, 49, 17, 2;
0, 0, 0, 0, 0, 0, 7, 50, 91, 64, 19, 2;
...

Cf. A029635, A029653, A178524.

G.f.: (1+x*y)/(1-x*y-x^2*y).
T(n,k) = T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000045(n+2), A026150(n+1), A108306(n), A164545(n), A188168(n+1) for x = 0, 1, 2, 3, 4, 5 respectively.

A241188 Triangle T(n,s) of Dynkin type D_n read by rows (n >= 2, 0 <= s <= n).

Original entry on oeis.org

1, 2, 1, 1, 3, 5, 5, 1, 4, 9, 16, 20, 1, 5, 14, 30, 55, 77, 1, 6, 20, 50, 105, 196, 294, 1, 7, 27, 77, 182, 378, 714, 1122, 1, 8, 35, 112, 294, 672, 1386, 2640, 4290, 1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445

Offset: 2

N. J. A. Sloane, Apr 24 2014

			Triangle begins:
1, 2, 1,
1, 3, 5, 5,
1, 4, 9, 16, 20,
1, 5, 14, 30, 55, 77,
1, 6, 20, 50, 105, 196, 294,
1, 7, 27, 77, 182, 378, 714, 1122,
1, 8, 35, 112, 294, 672, 1386, 2640, 4290,
1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445,
...

M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014.
M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, arXiv:1403.5827 [math.RT], 2014 and J. Int. Seq. 18 (2015) 15.10.6.

See A009766 for the case of type A.
See A059481 for the case of type B/C.
Diagonals give A029869, A051960, A029651, A051924. Row sums are also A051924.

f[t_, s_] := Binomial[t, s] (s + t)/t;
T[, 0] = 1; T[n, n_] := f[2 n - 2, n - 2]; T[n_, s_] := f[n + s - 2, s];
Table[T[n, s], {n, 2, 9}, {s, 0, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

T(n,s) = [n+s-2,s] for 0 <= s < n, T(n,n) = [2n-2,n-2], where [t,s] stands for binomial(t,s)*(s+t)/t.

cp's OEIS Frontend

A238160 A skewed version of triangular array A029653.

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