A394216 - OEIS

%I #7 Mar 17 2026 21:27:50

%S 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,

%T 700,560,1,42,346,2076,3449,6898,2868,1912,1,79,1264,4908,19632,22740,

%U 36384,11424,6528,1,252,2671,21368,52324,156972,135900,181200,44576,22288,1

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

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

%H Yury Belousov, <a href="http://ftp.pdmi.ras.ru/pub/publicat/znsl/v549/p049.pdf">Singular meanders</a>, Zap. Nauchn. Sem. POMI, 549 (2025), 49-64.

%H Yury Belousov, <a href="https://arxiv.org/abs/2112.10289">Prime Factorization of Meanders</a>, arXiv:2112.10289 [math.CO], 2025.

%H Yury Belousov, <a href="https://github.com/YuryBelousov/meander_factorization">C++ code for generating the sequence</a>, GitHub.

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

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

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

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

%e Triangle begins:

%e   1;

%e   1,  1;

%e   1,  4,  1;

%e   2,  7, 14,  1;

%e   3, 24, 36, 48, 1;

%e   ...

%t sOdd[x_, t_] := -2 x + 2 x/((1 - t)^2 - x^2);

%t sEven[x_, t_] := (1 - t)/((1 - t)^2 - x^2) - 1 - t;

%t iSer[n_, k_] := Module[{lam = 0, m = n - k},

%t   Do[lam = Expand@Normal@Series[sOdd[x + lam/2, t], {x, 0, m}, {t, 0, k}], {m + k + 2}];

%t   Expand@Normal@Series[lam + sEven[x + lam/2, t] + x + t + 1, {x, 0, m}, {t, 0, k}]];

%t T[n_, k_] := If[0 <= k <= n, Coefficient[Coefficient[iSer[n, k], x, n - k], t, k], 0];

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}]

%Y Column 0 is A007165.

%Y Main diagonal is A000012.

%Y Subdiagonal is A007070.

%Y The next diagonal is A181292.

%K nonn,tabl

%O 0,5

%A _Yury Belousov_, Mar 12 2026