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
Keywords
Examples
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 + ...
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.
Programs
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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 *)
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
Extensions
Corrected and extended by Vladeta Jovovic, Oct 24 2003
Comments