A256522 -id:A256522 - cp's OEIS frontend

A018893 Blasius sequence: from coefficients in expansion of solution to Blasius's equation in boundary layer theory.

1, 1, 11, 375, 27897, 3817137, 865874115, 303083960103, 155172279680289, 111431990979621729, 108511603921116483579, 139360142400556127213655, 230624017175131841824732233, 482197541715276031774659298833

Offset: 0

Stan Richardson (stan(AT)maths.ed.ac.uk)

nonn

Number of increasing trilabeled unordered trees. - Markus Kuba, Nov 18 2014

			A(x) = 1 + 1/6*x^3 + 11/720*x^6 + 25/24192*x^9 + 9299/159667200*x^12 + ...
G.f. = 1 + x + 11*x^3 + 375*x^4 + 27897*x^5 + 3817137*x^6 + ...

H. T. Davis: Introduction to Nonlinear Differential and Integral Equations (Dover 1962), page 403.

Vaclav Kotesovec, Table of n, a(n) for n = 0..180
Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Inaugural Dissertation, Georg-August-Universität Göttingen, Leipzig, 1907; see p. 8 (a(6) = c_6 and a(7) = c_7 are wrong in the dissertation) [USA access only].
Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37; see p. 8 (a(6) = c_6 has been corrected, while a(7) = c_7 was re-calculated incorrectly!).
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 (a(6) = c_6 has been corrected, while a(7) = c_7 was re-calculated incorrectly!).
Steven R. Finch, Prandtl-Blasius Flow. [Cached copy, with permission of the author]
W. H. Hager, Blasius: A life in research and education, Exp. Fluids 34(5) (2003), 566-571.
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.
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
Hans Salié, Über die Koeffizienten der Blasiusschen Reihen, Math. Nachr. 14 (1955), 241-248 (1956). [He generalizes the Blasius numbers.]
Wikipedia, Paul Richard Heinrich Blasius.

Cf. A002105, A256522.

a[0] = 1; a[k_] := a[k] = Sum[Binomial[3*k-1, 3*j]*a[j]*a[k-j-1], {j, 0, k-1}]; Table[a[k], {k, 0, 13}] (* Jean-François Alcover, Oct 28 2014 *)

E.g.f. A(x) satisfies (d^3/dx^3)log(A(x)) = A(x). - Vladeta Jovovic, Oct 24 2003
Lim_{n->infinity} (a(n)/(3*n+2)!)^(1/n) = 0.03269425181024... . - Vaclav Kotesovec, Oct 28 2014
T(z) = log(A(z)) satisfies T'''(z)=exp(T(z)), such that F(z)=T'(z) satisfies a Blasius type equation: F'''(z)-F(z)*F''(z)=0. - Markus Kuba, Nov 18 2014
a(n) = Sum_{v = 0..n-1} binomial(3*n-1, 3*v) * a(v) * a(n-1-v) for n >= 1 with a(0) = 1 (Blasius' recurrence). - Petros Hadjicostas, Aug 01 2019

Corrected and extended by Vladeta Jovovic, Oct 24 2003

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

Original entry on oeis.org

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

Petros Hadjicostas and Peter Luschny, Aug 06 2019

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.

			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|

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.]

Rows include A000108, A000142, A002105 (shifted), A018893.
Columns include A260878.
Cf. A256522 (Blasius constant), A260876.

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);

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 *)

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.

cp's OEIS Frontend

A018893 Blasius sequence: from coefficients in expansion of solution to Blasius's equation in boundary layer theory.

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