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A309522 Generalized Blasius numbers, square array read by ascending diagonals, A(n, k) for n, k >= 0. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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.

LINKS

Table of n, a(n) for n=0..54.

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.]

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.

EXAMPLE

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

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 *)

CROSSREFS

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

Columns include A260878.

Cf. A256522 (Blasius constant), A260876.

Sequence in context: A327722 A334548 A143364 * A336878 A305313 A159046

Adjacent sequences:  A309519 A309520 A309521 * A309523 A309524 A309525

KEYWORD

nonn,tabl

AUTHOR

Petros Hadjicostas and Peter Luschny, Aug 06 2019

STATUS

approved

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Last modified September 20 23:04 EDT 2021. Contains 347596 sequences. (Running on oeis4.)