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A049029 Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.

%I A049029 #73 Sep 04 2019 17:16:52
%S A049029 1,5,1,45,15,1,585,255,30,1,9945,5175,825,50,1,208845,123795,24150,
%T A049029 2025,75,1,5221125,3427515,775845,80850,4200,105,1,151412625,
%U A049029 108046575,27478710,3363045,219450,7770,140,1,4996616625,3824996175,1069801425
%N A049029 Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.
%C A049029 Previous name was: Triangle of numbers related to triangle A048882; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
%C A049029 a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quintic (5-ary) trees. Proof based on the a(n,m) recurrence. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - _Wolfdieter Lang_, Sep 14 2007
%C A049029 Also the Bell transform of A007696(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016
%H A049029 F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://dx.doi.org/10.1007/3-540-55251-0_2">Varieties of Increasing Trees</a>, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
%H A049029 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H A049029 T. Copeland, <a href="https://tcjpn.wordpress.com/2008/06/12/mathemagical-forests/">Mathemagical Forests</a>
%H A049029 T. Copeland, <a href="https://tcjpn.wordpress.com/2010/12/28/14/">Addendum to Mathemagical Forests</a>
%H A049029 T. Copeland, <a href="http://tcjpn.wordpress.com/2015/08/23/a-class-of-differential-operators-and-the-stirling-numbers/">A Class of Differential Operators and the Stirling Numbers</a>
%H A049029 M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) #09.8.3.
%H A049029 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A049029 W. Lang, <a href="/A049029/a049029.txt">First 10 rows</a>.
%H A049029 Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012
%H A049029 E. Neuwirth, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00373-3">Recursively defined combinatorial Functions: Extending Galton's board</a>, Discr. Maths. 239 (2001) 33-51.
%H A049029 Mathias Pétréolle, Alan D. Sokal, <a href="https://arxiv.org/abs/1907.02645">Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions</a>, arXiv:1907.02645 [math.CO], 2019.
%F A049029 a(n, m) = n!*A048882(n, m)/(m!*4^(n-m)); a(n+1, m) = (4*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1; E.g.f. of m-th column: ((-1+(1-4*x)^(-1/4))^m)/m!.
%F A049029 a(n, m) = sum(|A051142(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.
%F A049029 From _Peter Bala_, Nov 25 2011: (Start)
%F A049029 E.g.f.: G(x,t) = exp(t*A(x)) = 1+t*x+(5*t+t^2)*x^2/2!+(45*t+15*t^2+t^3)*x^3/3!+..., where A(x) = -1+(1-4*x)^(-1/4) satisfies the autonomous differential equation A'(x) = (1+A(x))^5.
%F A049029 The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-4*x)*dG/dx, from which follows the recurrence given above.
%F A049029 The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^5*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A035469 (D = (1+x)^4*d/dx).
%F A049029 (End)
%e A049029 Triangle starts:
%e A049029 {1};
%e A049029 {5,1};
%e A049029 {45,15,1};
%e A049029 {585,255,30,1};
%e A049029 {9945,5175,825,50,1};
%e A049029 ...
%p A049029 # The function BellMatrix is defined in A264428.
%p A049029 # Adds (1,0,0,0, ..) as column 0.
%p A049029 BellMatrix(n -> mul(4*k+1, k=0..n), 9); # _Peter Luschny_, Jan 28 2016
%t A049029 a[n_, m_] /; n >= m >= 1 := a[n, m] = (4(n-1) + m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover_, Jul 22 2011 *)
%t A049029 rows = 9;
%t A049029 a[n_, m_] := BellY[n, m, Table[Product[4k+1, {k, 0, j}], {j, 0, rows}]];
%t A049029 Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%Y A049029 a(n, m) := S2(5, n, m) is the fifth triangle of numbers in the sequence S2(1, n, m) := A008277(n, m) (Stirling 2nd kind), S2(2, n, m) := A008297(n, m) (Lah), S2(3, n, m) := A035342(n, m), S2(4, n, m) := A035469(n, m). a(n, 1)= A007696(n). A007559(n).
%Y A049029 Cf. A048882, A007696. Row sums: A049120(n), n >= 1.
%Y A049029 Cf. A094638
%K A049029 easy,nonn,tabl
%O A049029 1,2
%A A049029 _Wolfdieter Lang_
%E A049029 New name from _Peter Luschny_, Jan 30 2016