A394216 Triangle read by rows: T(n,k) is the number of iterated snakes with n intersections and k tangential intersections.

1, 1, 1, 1, 4, 1, 2, 7, 14, 1, 3, 24, 36, 48, 1, 8, 47, 188, 164, 164, 1, 14, 168, 453, 1208, 700, 560, 1, 42, 346, 2076, 3449, 6898, 2868, 1912, 1, 79, 1264, 4908, 19632, 22740, 36384, 11424, 6528, 1, 252, 2671, 21368, 52324, 156972, 135900, 181200, 44576, 22288, 1

Offset

0,5

Comments

An iterated snake is a type of singular meander that can be obtained from a snake by finitely many insertions of snakes.

Links

Table of n, a(n) for n=0..54.

Yury Belousov, Singular meanders, Zap. Nauchn. Sem. POMI, 549 (2025), 49-64.

Yury Belousov, Prime Factorization of Meanders, arXiv:2112.10289 [math.CO], 2025.

Yury Belousov, C++ code for generating the sequence, GitHub.

Formula

Let F(x,t) = Sum_{m,k>=0} T(m+k,k) x^m t^k.

Then U(x,t) := 1 + F(x,t) is the unique formal power series with U(0,0)=1 satisfying

(1 - t)*U^3 - (4*(1 - t)^2 - 2*(1 - t)*x + 1)*U^2 + (6*(1 - t) - x)*U - 2 = 0.

Hence T(n,k) = [x^(n-k) t^k] U(x,t).

Example

Triangle begins:
  1;
  1,  1;
  1,  4,  1;
  2,  7, 14,  1;
  3, 24, 36, 48, 1;
  ...

Mathematica

sOdd[x_, t_] := -2 x + 2 x/((1 - t)^2 - x^2);
sEven[x_, t_] := (1 - t)/((1 - t)^2 - x^2) - 1 - t;
iSer[n_, k_] := Module[{lam = 0, m = n - k},
  Do[lam = Expand@Normal@Series[sOdd[x + lam/2, t], {x, 0, m}, {t, 0, k}], {m + k + 2}];
  Expand@Normal@Series[lam + sEven[x + lam/2, t] + x + t + 1, {x, 0, m}, {t, 0, k}]];
T[n_, k_] := If[0 <= k <= n, Coefficient[Coefficient[iSer[n, k], x, n - k], t, k], 0];
Table[T[n, k], {n, 0, 8}, {k, 0, n}]

Crossrefs

Column 0 is A007165.

Main diagonal is A000012.

Subdiagonal is A007070.

The next diagonal is A181292.

Sequence in context: A243584 A084460 A216863 * A391919 A120578 A096249

Adjacent sequences: A394213 A394214 A394215 * A394217 A394218 A394219

Keyword

nonn,tabl

Author

Yury Belousov, Mar 12 2026

Status

approved