A341359 Square array T(m,n) read by antidiagonals, satisfying shifted Catalan recurrences: T(m,0) = 1 and T(m,n) = Sum_{k=0..n-1} T(m,k) * T(m,(n-1-k+m) mod n) for all n > 0.

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 5, 14, 1, 1, 2, 6, 15, 42, 1, 1, 2, 5, 26, 51, 132, 1, 1, 2, 5, 30, 142, 188, 429, 1, 1, 2, 6, 14, 305, 882, 731, 1430, 1, 1, 2, 5, 17, 210, 3955, 5910, 2950, 4862, 1, 1, 2, 5, 22, 50, 5894, 57855, 41610, 12235, 16796, 1, 1, 2, 6, 30, 65, 2550, 209146, 908880, 303390, 51822, 58786

Offset

0,6

Comments

Each column is periodic, and the period of column n divides A003418(n).

Links

Table of n, a(n) for n=0..77.

T. Amdeberhan et al. Sequences generated by sum & product of terms (with rotating indices): combinatorial?, 2021.

Formula

G.f. for row m: p_m(x) + x^(m-1)/2 * ( 1 + sqrt((1 -(4*T(m,m)+1)*x)/(1-x)) ), where p_m(x) = Sum_{n=0..m-1} T(m,n) * x^n.

Example

Rows of the array:
m=0: 1, 1, 2, 5, 14,  42,  132,    429,     1430, ...
m=1: 1, 1, 2, 5, 15,  51,  188,    731,     2950, ...
m=2: 1, 1, 2, 6, 26, 142,  882,   5910,    41610, ...
m=3: 1, 1, 2, 5, 30, 305, 3955,  57855,   908880, ...
m=4: 1, 1, 2, 5, 14, 210, 5894, 209146,  8331582, ...
m=5: 1, 1, 2, 6, 17,  50, 2550, 255050, 32007550, ...
...

Mathematica

T[m_, 0] := 1; T[m_, n_] := T[m, n] = Sum[T[m, k] * T[m, Mod[n - 1 - k + m, n]], {k, 0, n - 1}]; Table[T[m - n, n], {m, 0, 11}, {n, 0, m}] // Flatten (* Amiram Eldar, Feb 09 2021 *)

Crossrefs

Rows: A000108 (m=0), A181768 (m=1).

Columns: A000012 (n=0 and n=1), A007395 (n=2).

Cf. A007317, A162326, A082298,A226392, A341360 (diagonal).

Sequence in context: A135723 A377442 A125311 * A127568 A263791 A377441

Adjacent sequences: A341356 A341357 A341358 * A341360 A341361 A341362

Keyword

nonn,tabl

Author

Max Alekseyev, Feb 09 2021

Status

approved