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A309522 Generalized Blasius numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.

%I A309522 #50 Jul 07 2024 09:28:44
%S A309522 1,1,1,1,1,2,1,1,2,5,1,1,4,6,14,1,1,11,34,24,42,1,1,36,375,496,120,
%T A309522 132,1,1,127,6306,27897,11056,720,429,1,1,463,129256,3156336,3817137,
%U A309522 349504,5040,1430,1,1,1717,2877883,514334274,3501788976,865874115,14873104,40320,4862
%N A309522 Generalized Blasius numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.
%C A309522 The generalized Blasius o.d.e. of order n whose infinite series solution involves row n of this square array appears in Salié (1955). Rows n = 2 and n = 3 of this array appear in Kuba and Panholzer (2014, 2016), who give combinatorial interpretations to the numbers in those two rows.
%C A309522 Eq. (22) in Kuba and Panholzer (2014, p. 23) and Eq. (5) in Kuba and Panholzer (2016, p. 233) are general o.d.e.s for generating infinite sequences of numbers with some combinatorial properties. Even though they bear some similarity to Salié's general o.d.e., it is not clear whether either one can be used to give combinatorial interpretation to the numbers in rows n >= 4 of the current square array.
%H A309522 Heinrich Blasius, <a href="https://iris.univ-lille.fr/handle/1908/2024">Grenzschichten in Flüssigkeiten mit kleiner Reibung</a>, Z. Math. u. Physik 56 (1908), 1-37; see p. 8. [This article was based on his PhD thesis. He corrected c_6 = A(n=3, k=6) but his "correction" of c_7 = A(n=3, k=7) was not right!]
%H A309522 Heinrich Blasius, <a href="http://naca.central.cranfield.ac.uk/reports/1950/naca-tm-1256.pdf">Grenzschichten in Flüssigkeiten mit kleiner Reibung</a>, Z. Math. u. Physik 56 (1908), 1-37 [English translation by J. Vanier on behalf of the National Advisory Committee for Aeronautics (NACA), 1950]; see p. 8. [This is a translation of Blasius' article. The value of c_6 = A(n=3, k=6) was corrected in the article and the translation, but the "correction" for c_7 = A(n=3, k=7) in both documents is wrong.]
%H A309522 Markus Kuba and Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 2014.
%H A309522 Markus Kuba and Alois Panholzer, <a href="https://doi.org/10.1016/j.disc.2015.08.010">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, Discrete Mathematics 339(1) (2016), 227-254.
%H A309522 Hans Salié, <a href="https://doi.org/10.1002/mana.19550140405"> Über die Koeffizienten der Blasiusschen Reihen</a>, Math. Nachr. 14 (1955), 241-248 (1956). [In the article the array is denoted by c^{(n)}_v for n, v >= 1. We have A(n, k) = c^{(n)}_{k+1} for n >= 1 and k >= 0. The Catalan numbers (row n = 0 for A(n, k)) do not appear in Salié's article.]
%F A309522 A(n, k) = Sum_{v=0..k-1} binomial(n*k-1, n*v)*A(n, v)*A(n, k-1-v) for k > 0 and A(n, 0) = 1.
%F A309522 A(n, 2) = A260876(n, 2) = binomial(2*n - 1, n) + 1 for n >= 0.
%F A309522 A(n, 3) = A260876(n, 2) + A260876(n, 3) - 1 = (binomial(3*n - 1, 2*n) + 1) * (binomial(2*n - 1, n) + 1) + binomial(3*n - 1, n) for n >= 1.
%e A309522 Table A(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e A309522 [0] 1, 1,  2,    5,      14,         42,           132, ...  A000108
%e A309522 [1] 1, 1,  2,    6,      24,        120,           720, ...  A000142
%e A309522 [2] 1, 1,  4,   34,     496,      11056,        349504, ...  A002105
%e A309522 [3] 1, 1, 11,  375,   27897,    3817137,     865874115, ...  A018893
%e A309522 [4] 1, 1, 36, 6306, 3156336, 3501788976, 7425169747776, ...
%e A309522      A260878|
%p A309522 A := proc(n, k) option remember; if k = 0 then 1 else
%p A309522 add(binomial(n*k-1, n*v)*A(n, v)*A(n, k-1-v), v=0..k-1) fi end:
%p A309522 seq(seq(A(n-k, k), k=0..n), n=0..9);
%t A309522 A[n_, k_] := A[n, k] = If[k == 0, 1, Sum[Binomial[n*k - 1, n*v]*A[n, v]* A[n, k - 1 - v], {v, 0, k - 1}]];
%t A309522 Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 26 2019, from Maple *)
%Y A309522 Rows include A000108, A000142, A002105 (shifted), A018893.
%Y A309522 Columns include A260878.
%Y A309522 Cf. A256522 (Blasius constant), A260876.
%K A309522 nonn,tabl
%O A309522 0,6
%A A309522 _Petros Hadjicostas_ and _Peter Luschny_, Aug 06 2019