A085478 -id:A085478 - cp's OEIS frontend

A373547 Triangle read by rows: T(n,k) = 4^k*binomial(n+k, n-k) with 0 <= k <= n.

1, 1, 4, 1, 12, 16, 1, 24, 80, 64, 1, 40, 240, 448, 256, 1, 60, 560, 1792, 2304, 1024, 1, 84, 1120, 5376, 11520, 11264, 4096, 1, 112, 2016, 13440, 42240, 67584, 53248, 16384, 1, 144, 3360, 29568, 126720, 292864, 372736, 245760, 65536, 1, 180, 5280, 59136, 329472, 1025024, 1863680, 1966080, 1114112, 262144

Offset: 0

Stefano Spezia, Jun 09 2024

T(n,k) is the number of occurrences of the periodic substring (01)^k in the periodic string (0011)^n (see Proposition 4.3 at page 6 in Fang).
The word (w_1, w_2, ..., w_r)^m is defined as the word obtained by concatenating (w_1, w_2, ..., w_r) m times.
A word w' = (w'1, w'_2, ..., w'_s) is said be a subword of a given word w = (w_1, w_2, ..., w_r), if there is some set P = {p_1 < ... < p_s} of integers from 1 to r satisfying w{p_j} = w'_j for all 1 <= j <= s, and we call the set P an occurrence of w' in w (see Preliminaries section at pp. 2-3 in Fang).

			The triangle begins as:
  1;
  1,  4;
  1, 12,   16;
  1, 24,   80,   64;
  1, 40,  240,  448,   256;
  1, 60,  560, 1792,  2304,  1024;
  1, 84, 1120, 5376, 11520, 11264, 4096;
  ...
T(2,1) = 12 since there are 12 occurrences of (01)^1 = 01 in (0011)^2 = 00110011: {1, 3}, {1, 4}, {1, 7}, {1, 8}, {2, 3}, {2, 4}, {2, 7}, {2, 8}, {5, 7}, {5, 8}, {6, 7}, {6, 8}.

Wenjie Fang, Maximal number of subword occurrences in a word, arXiv:2406.02971 [math.CO], 2024.

Cf. A000012 (k=0), A000302 (diagonal), A001653 (row sums), A046092 (k=1), A046717, A085478, A130810, A130812, A373628.

T[n_,k_]:=4^k Binomial[n+k,n-k]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* or *)
T[n_,k_]:=SeriesCoefficient[(1-x)/((1-x)^2-4x y),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten

G.f.: (1 - x)/((1 - x)^2 - 4*x*y).
T(n,k) = A000302(k)*A085478(n,k).
Sum_{k=0..n} T(n-k,k) = A046717(n).
T(n,2) = A130810(n+2).
T(n,3) = A130812(n+3).

A373628 Triangle read by rows: T(n,k) is the number of occurrences of the periodic substring (0011)^k in the periodic string (000111)^n.

Original entry on oeis.org

1, 1, 9, 1, 81, 81, 1, 351, 1377, 729, 1, 1035, 11421, 18225, 6561, 1, 2430, 62613, 223803, 216513, 59049, 1, 4914, 259119, 1813023, 3523257, 2421009, 531441, 1, 8946, 874071, 10978740, 37850409, 49069719, 26040609, 4782969, 1, 15066, 2525499, 53362800, 303255981, 657274419, 631883349, 272629233, 43046721

Offset: 0

Stefano Spezia, Jun 11 2024

The word (w_1, w_2, ..., w_r)^m is defined as the word obtained by concatenating (w_1, w_2, ..., w_r) m times.

A word w' = (w'1, w'_2, ..., w'_s) is said be a subword of a given word w = (w_1, w_2, ..., w_r), if there is some set P = {p_1 < ... < p_s} of integers from 1 to r satisfying w{p_j} = w'_j for all 1 <= j <= s, and we call the set P an occurrence of w' in w (see Preliminaries section at pp. 2-3 in Fang).

			The triangle begins as:
  1;
  1,    9;
  1,   81,    81;
  1,  351,  1377,    729;
  1, 1035, 11421,  18225,   6561;
  1, 2430, 62613, 223803, 216513, 59049;
  ...
T(1,1) = 9 since there are 9 occurrences of (0011)^1 = 0011 in (000111)^1 = 000111: {1, 2, 4, 5}, {1, 2, 4, 6}, {1, 2, 5, 6}, {1, 3, 4, 5}, {1, 3, 4, 6}, {1, 3, 5, 6}, {2, 3, 4, 5}, {2, 3, 4, 6}, {2, 3, 5, 6}.

Wenjie Fang, Maximal number of subword occurrences in a word, arXiv:2406.02971 [math.CO], 2024. See Proposition 4.10 at page 9 in Fang.

Cf. A000012 (k=0), A001019 (diagonal), A085478, A373547.

T[n_, k_]:=SeriesCoefficient[(1-x)^3/((1-x)^4-9x(1+2x)^2y), {x, 0, n}, {y, 0, k}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}]//Flatten

G.f.: (1 - x)^3/((1 - x)^4 - 9*x*(1 + 2*x)^2*y).

A177040 Irregular triangle t(n,m) = binomial(m+1,n-m) read by rows floor((n+1)/2) <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 4, 1, 6, 5, 1, 4, 10, 6, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, 21, 70, 84, 45, 11, 1, 7, 56, 126, 120, 55, 12, 1, 28, 126, 210, 165, 66, 13, 1, 8, 84, 252, 330, 220, 78, 14, 1, 36, 210, 462, 495, 286, 91, 15, 1

Offset: 0

Roger L. Bagula, May 01 2010

Row sums are in A052952.

Contains the right half of each row of A030528. - R. J. Mathar, May 19 2013

			1;
1;
2, 1;
3, 1;
3, 4, 1;
6, 5, 1;
4, 10, 6, 1;
10, 15, 7, 1;
5, 20, 21, 8, 1;
15, 35, 28, 9, 1;
6, 35, 56, 36, 10, 1;
21, 70, 84, 45, 11, 1;
7, 56, 126, 120, 55, 12, 1;
28, 126, 210, 165, 66, 13, 1;
8, 84, 252, 330, 220, 78, 14, 1;
36, 210, 462, 495, 286, 91, 15, 1;

Cf. A180987 (read diagonally downwards), A098925, A026729, A085478, A165253

t[n_, m_] := Binomial[m + 1, n - m];
Table[Table[t[n, m], {m, Floor[(n + 1)/2], n}], {n, 0, 15}];
Flatten[%]

T(m,n)=binomial(n+1,m-n) \\ Charles R Greathouse IV, May 19 2013

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A373547 Triangle read by rows: T(n,k) = 4^k*binomial(n+k, n-k) with 0 <= k <= n.

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A373628 Triangle read by rows: T(n,k) is the number of occurrences of the periodic substring (0011)^k in the periodic string (000111)^n.

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A177040 Irregular triangle t(n,m) = binomial(m+1,n-m) read by rows floor((n+1)/2) <= m <= n.

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