A131568 -id:A131568 - cp's OEIS frontend

A362030 Irregular triangle read by rows where row n contains the balanced binary words of length 2n interpreted as binary numbers.

1, 2, 3, 5, 6, 9, 10, 12, 7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 135

Offset: 1

Louis Conover, Apr 05 2023

Within a row, strings are ordered lexicographically, which means the resulting values are ordered numerically.
This is from an idea of David Lovler, which he calls "zigzags". It is a rearrangement of A072601. A072603 lists all the numbers that are not in this sequence. A000984 gives the number of coin flip sequences of length 2,4,6, etc.
Not a permutation of the integers. E.g. 8 never occurs. When there are more 0's than 1's, adding 0's doesn't bring it to balance. - Kevin Ryde, Aug 31 2023

			The first few terms written as binary words with leading 0's: 01, 10, 0011, 0101, 0110, 1001, 1010, 1100, 000111, 001011, 001101, 001110, ... (cf. A368804).
Triangle T(n,k) begins:
   1,  2;
   3,  5,  6,  9, 10, 12;
   7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, ...;
  15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, ...;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..17576 (rows 1..8 of the triangle, flattened).

Columns k=1-2 give: A000225, A083329.
Row sums give A131568.
Main diagonal gives A036563(n+1).
Cf. A000984 (row lengths), A072601, A072603, A368804 (binary).

T:= n-> sort(map(Bits[Join], combinat[permute]([0$n, 1$n])))[]:
seq(T(n), n=1..4);  # Alois P. Heinz, Apr 13 2023

T[n_] := Sort[FromDigits[#, 2] & /@ Permutations[Join[ConstantArray[0, n], ConstantArray[1, n]]]]; Flatten[Table[T[n], {n, 1, 4}]][[1 ;; 64]] (* Robert P. P. McKone, Aug 29 2023 *)

A356117 T(n, k) = [x^k] (1/2 - x)^(-n) - (1 - x)^(-n).

Original entry on oeis.org

0, 1, 3, 3, 14, 45, 7, 45, 186, 630, 15, 124, 630, 2540, 8925, 31, 315, 1905, 8925, 35770, 128898, 63, 762, 5355, 28616, 128898, 515844, 1891890, 127, 1785, 14308, 85932, 429870, 1891890, 7568484, 28113228, 255, 4088, 36828, 245640, 1351350, 6487272, 28113228, 112456344, 421717725

Offset: 0

Peter Luschny, Aug 22 2022

			Triangle T(n, k) starts:
[0]   0;
[1]   1,    3;
[2]   3,   14,    45;
[3]   7,   45,   186,    630;
[4]  15,  124,   630,   2540,    8925;
[5]  31,  315,  1905,   8925,   35770,  128898;
[6]  63,  762,  5355,  28616,  128898,  515844,  1891890;
[7] 127, 1785, 14308,  85932,  429870, 1891890,  7568484,  28113228;
[8] 255, 4088, 36828, 245640, 1351350, 6487272, 28113228, 112456344, 421717725;

Cf. A000225 (column 0), A059672 (column 1), A059937 (column 2), A131568 (main diagonal), A134346, A327318.

ser := series((1/2 - x)^(-n) - (1 - x)^(-n), x, 20):
seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9);

row[n_] := CoefficientList[Series[(1/2 - x)^(-n) - (1 - x)^(-n), {x, 0, n}], x]; row[0] = {0}; Table[row[n], {n, 0, 8}] // Flatten (* Amiram Eldar, Aug 22 2022 *)

T(n, k) = (2^(n+k) - 1) * binomial(n+k-1, k). - John Keith, Aug 23 2022

cp's OEIS Frontend

A362030 Irregular triangle read by rows where row n contains the balanced binary words of length 2n interpreted as binary numbers.

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