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
Examples
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, ... ...
Links
- T. Amdeberhan et al. Sequences generated by sum & product of terms (with rotating indices): combinatorial?, 2021.
Crossrefs
Programs
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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 *)
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.
Comments