A004145 -id:A004145 - cp's OEIS frontend

A202848 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of even length (n>=0, k>=0).

1, 1, 1, 2, 4, 7, 1, 14, 3, 31, 6, 66, 16, 141, 44, 313, 107, 3, 702, 262, 14, 1577, 663, 43, 3581, 1654, 138, 8207, 4091, 436, 1, 18903, 10178, 1275, 16, 43770, 25339, 3638, 85, 101903, 62952, 10316, 331, 238282, 156495, 28743, 1228, 559322, 389374, 78979, 4320, 9

Offset: 0

Emeric Deutsch, Dec 26 2011

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
Sum of entries in row n is A004148 (the secondary structure numbers).
Sum(k*T(n,k), k>=0) = A202846(n-2).
T(n,0) = A202849(n).

			Row 5 is 7,1: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv; ABVBA; the last one has 1 stack of length 2.
Triangle starts:
1;
1;
1;
2;
4;
7,1;
14,3;
31,6;

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A004145, A202846, A023427, A202849,

f := (z^2+t*z^4)/(1-z^4): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 19 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form

G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (z^2 + t*z^4)/(1-z^4).

The multivariate g.f. H(z, t[1], t[2], ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + (f/(1 + f))H(H-1), where f = t[1]z^2 + t[2]z^4 + t[3]z^6 + ... .

A268408 Triangle T(d,v) read by rows: the number of hyper-tetrahedra with volume v/d! defined by selecting d+1 vertices of the d-dimensional unit-hypercube.

Original entry on oeis.org

0, 1, 0, 4, 12, 56, 2, 1360, 2672, 320, 16, 350000, 431232, 107904, 12864, 3872, 320, 255036992, 234667968, 98251776, 19523136, 10633728, 1615552, 1182720, 163520, 127360, 13440

Offset: 1

R. J. Mathar, Feb 04 2016

The unit hypercube in dimension d has 2^d vertices, conveniently expressed by their Cartesian coordinates as binary vectors of length d of 0's and 1's. Hyper-tetrahedra (simplices) are defined by selecting a subset of 1+d of them. The (signed) volume V of a tetrahedron is the determinant of the d vectors of the edges divided by d!. (The volume may be zero if some edges in the tetrahedron are linearly dependent.) The triangle T(d,v) is a histogram of all A136465(d+1) tetrahedra classified by absolute (unsigned) volume V=v/d!.

The number of non-flat simplices (row sums without the leftmost column) are tabulated by Brandts et al. (Table 1, column beta_n). - R. J. Mathar, Feb 06 2016

			In d=2, 4 tetrahedra (triangles) are defined by taking subsets of d+1=3 vertices out of the 2^2=4 vertices of the unit square. Each of them has the same volume (area) 1/2!, so T(d=2,v=1)=4.
In d=3, 12 = T(d=3,v=0) tetrahedra with zero volume are defined by taking subsets of d+1=4 vertices out of the 2^3=8 vertices of the unit cube. These are the cases of taking any 4 vertices on a common face. (There are 6 faces and two different edge sets for each of them; one with edges along the cube's edges, and one with edges along the face diagonals.)
The triangle starts in row d=1 as follows:
0 1;
0  4;
12  56  2;
1360  2672  320  16 ;
350000  431232  107904  12864  3872  320;

J. Brandts, Sander Dijkhuis et al, There are only two nonobtuse binary triangulations of the unit n-cube, arXiv:1209.3875 and Comput. Geometry 46 (2013) 286

Cf. A136465 (row sums), A003432 (maximum column index), A004145 (column v=0).

cp's OEIS Frontend

A202848 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of even length (n>=0, k>=0).

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