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Blasius sequence: from coefficients in expansion of solution to Blasius's equation in boundary layer theory.
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%I #86 Jul 16 2021 08:25:38

%S 1,1,11,375,27897,3817137,865874115,303083960103,155172279680289,

%T 111431990979621729,108511603921116483579,139360142400556127213655,

%U 230624017175131841824732233,482197541715276031774659298833

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

%C Number of increasing trilabeled unordered trees. - _Markus Kuba_, Nov 18 2014

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

%H Vaclav Kotesovec, <a href="/A018893/b018893.txt">Table of n, a(n) for n = 0..180</a>

%H Heinrich Blasius, <a href="https://play.google.com/store/books/details?id=1lFIAQAAMAAJ&amp;rdid=book-1lFIAQAAMAAJ&amp;rdot=1">Grenzschichten in Flüssigkeiten mit kleiner Reibung</a>, 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].

%H Heinrich Blasius, <a href="https://iris.univ-lille.fr/handle/1908/2024">Grenzschichten in Flüssigkeiten mit kleiner Reibung</a>, 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!).

%H Heinrich Blasius, <a href="http://naca.central.cranfield.ac.uk/reports/1950/naca-tm-1256.pdf">Grenzschichten in Flüssigkeiten mit kleiner Reibung</a>, 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!).

%H Steven R. Finch, <a href="/A018893/a018893.pdf">Prandtl-Blasius Flow</a>. [Cached copy, with permission of the author]

%H W. H. Hager, <a href="https://doi.org/10.1007/s00348-002-0582-9">Blasius: A life in research and education</a>, Exp. Fluids 34(5) (2003), 566-571.

%H Markus Kuba and Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 2014.

%H Markus Kuba and Alois Panholzer, <a href="https://doi.org/10.1016/j.disc.2015.08.010">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, Discrete Mathematics 339(1) (2016), 227-254.

%H Ronald Orozco López, <a href="https://www.researchgate.net/publication/350397609_Solution_of_the_Differential_Equation_ykeay_Special_Values_of_Bell_Polynomials_and_ka-Autonomous_Coefficients">Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients</a>, Universidad de los Andes (Colombia 2021).

%H Hans Salié, <a href="https://doi.org/10.1002/mana.19550140405"> Über die Koeffizienten der Blasiusschen Reihen</a>, Math. Nachr. 14 (1955), 241-248 (1956). [He generalizes the Blasius numbers.]

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Paul_Richard_Heinrich_Blasius">Paul Richard Heinrich Blasius</a>.

%F E.g.f. A(x) satisfies (d^3/dx^3)log(A(x)) = A(x). - _Vladeta Jovovic_, Oct 24 2003

%F Lim_{n->infinity} (a(n)/(3*n+2)!)^(1/n) = 0.03269425181024... . - _Vaclav Kotesovec_, Oct 28 2014

%F 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

%F 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

%e A(x) = 1 + 1/6*x^3 + 11/720*x^6 + 25/24192*x^9 + 9299/159667200*x^12 + ...

%e G.f. = 1 + x + 11*x^3 + 375*x^4 + 27897*x^5 + 3817137*x^6 + ...

%t 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 *)

%Y Cf. A002105, A256522.

%K nonn

%O 0,3

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

%E Corrected and extended by _Vladeta Jovovic_, Oct 24 2003