cp's OEIS Frontend

T(h,k) is the number of classes of aperiodic binary words of k ones and h-k zeros; words u,v are in the same class if v is a cyclic permutation of u (e.g., u=111000, v=110001) and a word is aperiodic if not a juxtaposition of 2 or more identical subwords.

T(2n, n), T(2n+1, n), T(n, 3) match A022553, A000108, A001840, respectively. Row sums match A001037.

From R. J. Mathar, Jul 31 2008: (Start)

This triangle may also be regarded as the square array A(r,n), the n-th term of the r-th Witt transform of the all-1 sequence, r >= 1, n >= 0, read by antidiagonals:

This array begins as follows:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63

0 1 2 5 8 14 20 30 40 55 70 91 112 140 168 204 240 285 330

0 1 3 7 14 25 42 66 99 143 200 273 364 476 612 775 969 1197 1463

0 1 3 9 20 42 75 132 212 333 497 728 1026 1428 1932 2583 3384 4389 5598

0 1 4 12 30 66 132 245 429 715 1144 1768 2652 3876 5537 7752 10659 14421 19228

0 1 4 15 40 99 212 429 800 1430 2424 3978 6288 9690 14520 21318 30624 43263 60060

0 1 5 18 55 143 333 715 1430 2700 4862 8398 13995 22610 35530 54477 81719 120175

0 1 5 22 70 200 497 1144 2424 4862 9225 16796 29372 49742 81686 130750 204248

0 1 6 26 91 273 728 1768 3978 8398 16796 32065 58786 104006 178296 297160 482885

0 1 6 30 112 364 1026 2652 6288 13995 29372 58786 112632 208012 371384 643842

0 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400023 742900 1337220

0 1 7 40 168 612 1932 5537 14520 35530 81686 178296 371384 742900 1432613 2674440

...

It is essentially symmetric: A(r,r+i) = A(r,r-i+1).

Some of the diagonals are:

A(r,r+1): A000108

A(r,r): A022553

A(r,r-1): A000108

A(r,r+2): A000150

A(r,r+3): A050181

A(r,r+4): A050182

A(r,r+5): A050183

A(r,r-2): A000150 (End)

Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} A054533(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = T(n + k, k). - Petros Hadjicostas, Jul 09 2019

Number of ways to distribute n pairs of parentheses into 2 distinct patterns where each pattern represents a Catalan ordering (A000108), and each pattern must contain at least one pair of parentheses.

If one of the groups is allowed to have no parentheses, we arrive at A000150 (with a different offset).

Analog of A216785 with Catalan number replacing connected graph counts.

From Petros Hadjicostas, Jul 27 2020: (Start)

It is proved in A050182 that A050182(n) = 1/(2*n + 4)*(binomial(2*n + 4, n) - [(n mod 2) == 0]*binomial(n + 2, n/2)).

Let C(x) = A(x) + 1 = Sum_{n >= 0} c(n)*x^n be the g.f. of the Catalan numbers A000108. Then C(x)^2 = (C(x) - 1)/x. Then (A(x) + 1)^2 = A(x)/x, and thus, A(x)^2 = -2*A(x) - 1 + A(x)/x. Thus, (A(x)^2 - A(x^2))/2 = (-2*A(x) - 1 + A(x)/x - A(x^2))/2.

Substituting A(x) = Sum_{n >= 1} c(n)*x^n in the above expression, we get (after some algebra) that a(n) = (-2*c(n) + c(n+1) - [(n mod 2) == 0]*c(n/2))/2 for n >= 1. It is then easy to prove that a(n) = 2*A050182(n-2) = (1/n)*(binomial(2*n, n-2) - [(n mod 2) == 0]*binomial(n, (n/2) - 1)) for n >= 2, thus proving the conjecture below. (End)