A391786 Triangle read by rows: T(n, k) = number of reversible strings with n black beads and k white beads, 0 <= k <= n.

1, 1, 1, 1, 2, 3, 1, 2, 6, 7, 1, 3, 9, 19, 23, 1, 3, 12, 28, 66, 71, 1, 4, 16, 44, 110, 236, 252, 1, 4, 20, 60, 170, 396, 868, 890, 1, 5, 25, 85, 255, 651, 1519, 3235, 3299, 1, 5, 30, 110, 365, 1001, 2520, 5720, 12190, 12283, 1, 6, 36, 146, 511, 1512, 4032, 9752, 21942, 46252, 46508

Offset

0,5

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Formula

For k < n, T(n, k) = (binomial(n+k, n) + S1(n, k)) / 2 where S1(n, k) is 0 if both n and k are odd, and S1(n, k) = binomial(floor((n+k)/2), floor(k/2)) otherwise.

T(n, n) = A045723(n) = (binomial(2*n, n)/2 + A027306(n)) / 2.

(S1(n, k) and A027306(n) are the number of symmetrical (achiral) strings.)

Example

Triangle T(n, k) begins:
  1;
  1, 1;
  1, 2,  3;
  1, 2,  6,  7;
  1, 3,  9, 19, 23;
  1, 3, 12, 28, 66, 71;
  ...
T(4, 0) = 1: nnnn.
T(4, 1) = 3: nnnnk, nnnkn, nnknn.
T(4, 2) = 9: nnnnkk, nnnknk, nnnkkn, nnknnk, nnknkn, nknnnk, nnkknn, nknnkn, knnnnk.
T(4, 3) = 19: nnnnkkk, knnnnkk, nnnknkk, nnnkknk, nnnkkkn, knnnknk, knnnkkn, kknnnkn, nnknnkk, nnkknnk, nnknknk, nnknkkn, nnkknkn, knnknkn, nknnknk, nknnkkn, nnkkknn, knnknnk, nknknkn.
T(4, 4) = 23: nnnkkkkn, nnnkkknk, nnnkknkk, nnkkknnk, nnknkkkn, nnkkknkn, nknnkkkn, nnknkknk, nnkknknk, nkknnknk, nnkknkkn, nkknknkn, nnnnkkkk, nnkkkknn, nnnknkkk, nkkknnnk, nnkknnkk, nnknknkk, nknnkknk, nkknknnk, nknknknk, nkknnkkn, nknkknkn.

Mathematica

A391786[n_, k_] := If[n == k, (2^n + Binomial[2*n, n] + 2*Binomial[n-1, (n-2)/2]*Mod[n+1, 2])/4, (Binomial[n+k, n] + If[AllTrue[{n, k}, OddQ], 0, Binomial[Quotient[n+k, 2], Quotient[k, 2]]])/2];
Table[A391786[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 11 2026 *)

Prog

(PARI) A391786(n, k)={if(k<n, (binomial(n+k, n) + if(n%2==0 || k%2==0, binomial((n+k)\2, k\2)))/2, A045723(n))} \\ M. F. Hasler, Feb 09 2026
/* then, e.g.: concat( [[A391786(n, k)|k<-[0..n]]|n<-[0..10]] ) */

Crossrefs

Antidiagonal sums give A005418.

Main diagonal is A045723.

2nd diagonal is A005654.

Cf. A027306.

Sequence in context: A109091 A138507 A209579 * A205699 A109200 A158909

Adjacent sequences: A391783 A391784 A391785 * A391787 A391788 A391789

Keyword

nonn,tabl

Author

Stephen G Taylor, Dec 20 2025

Status

approved