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

0,3

Comments

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

References

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

Links

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.

Formula

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

Example

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 + ...

Mathematica

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 *)

Crossrefs

Cf. A002105, A256522.

Sequence in context: A024149 A353934 A333466 * A051862 A006698 A348353

Adjacent sequences: A018890 A018891 A018892 * A018894 A018895 A018896

Keyword

nonn

Author

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

Extensions

Corrected and extended by Vladeta Jovovic, Oct 24 2003

Status

approved