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

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, 132, 1, 1, 127, 6306, 27897, 11056, 720, 429, 1, 1, 463, 129256, 3156336, 3817137, 349504, 5040, 1430, 1, 1, 1717, 2877883, 514334274, 3501788976, 865874115, 14873104, 40320, 4862

Offset: 0

Author

Petros Hadjicostas and Peter Luschny, Aug 06 2019

Keywords

• nonn
• tabl

Comments

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.

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.

Examples

			Table A(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
[0] 1, 1,  2,    5,      14,         42,           132, ...  A000108
[1] 1, 1,  2,    6,      24,        120,           720, ...  A000142
[2] 1, 1,  4,   34,     496,      11056,        349504, ...  A002105
[3] 1, 1, 11,  375,   27897,    3817137,     865874115, ...  A018893
[4] 1, 1, 36, 6306, 3156336, 3501788976, 7425169747776, ...
     A260878|
		

Links

• Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, 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!]
• Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, 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.]
• Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
• Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, Discrete Mathematics 339(1) (2016), 227-254.
• Hans Salié, Über die Koeffizienten der Blasiusschen Reihen, 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.]

Crossrefs

Rows include A000108, A000142, A002105 (shifted), A018893.

Columns include A260878.

Cf. A256522 (Blasius constant), A260876.

Programs

Maple

A := proc(n, k) option remember; if k = 0 then 1 else
add(binomial(n*k-1, n*v)*A(n, v)*A(n, k-1-v), v=0..k-1) fi end:
seq(seq(A(n-k, k), k=0..n), n=0..9);

Mathematica

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}]];
Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *)

Formula

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.

A(n, 2) = A260876(n, 2) = binomial(2*n - 1, n) + 1 for n >= 0.

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.