|
%I #130 Mar 05 2024 06:06:08
%S 1,3,1,15,9,1,105,87,18,1,945,975,285,30,1,10395,12645,4680,705,45,1,
%T 135135,187425,82845,15960,1470,63,1,2027025,3133935,1595790,370125,
%U 43890,2730,84,1,34459425,58437855,33453945,8998290
%N The convolution matrix of the double factorial of odd numbers (A001147).
%C Previous name was: A triangle of numbers related to the triangle A035324; generalization of Stirling numbers of second kind A008277 and Lah numbers A008297.
%C If one replaces in the recurrence the '2' by '0', resp. '1', one obtains the Lah-number, resp. Stirling-number of 2nd kind, triangle A008297, resp. A008277.
%C The product of two lower triangular Jabotinsky matrices (see A039692 for the Knuth 1992 reference) is again such a Jabotinsky matrix: J(n,m) = Sum_{j=m..n} J1(n,j)*J2(j,m). The e.g.f.s of the first columns of these triangular matrices are composed in the reversed order: f(x)=f2(f1(x)). With f1(x)=-(log(1-2*x))/2 for J1(n,m)=|A039683(n,m)| and f2(x)=exp(x)-1 for J2(n,m)=A008277(n,m) one has therefore f2(f1(x))=1/sqrt(1-2*x) - 1 = f(x) for J(n,m)=a(n,m). This proves the matrix product given below. The m-th column of a Jabotinsky matrix J(n,m) has e.g.f. (f(x)^m)/m!, m>=1.
%C a(n,m) gives the number of forests with m rooted ordered trees with n non-root vertices labeled in an organic way. Organic labeling means that the vertex labels along the (unique) path from the root with label 0 to any leaf (non-root vertex of degree 1) is increasing. Proof: first for m=1 then for m>=2 using the recurrence relation for a(n,m) given below. - _Wolfdieter Lang_, Aug 07 2007
%C Also the Bell transform of A001147(n+1) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 19 2016
%H Reinhard Zumkeller, <a href="/A035342/b035342.txt">Rows n = 1..125 of triangle, flattened</a>
%H Peter Bala, <a href="/A035342/a035342_Bala.txt">Generalized Dobinski formulas</a>
%H J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, <a href="https://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013-2014.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>, arXiv:quant-ph/0212072, 2002.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H Richell O. Celeste, Roberto B. Corcino and Ken Joffaniel M. Gonzales. <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Celeste/celeste3.html"> Two Approaches to Normal Order Coefficients</a>. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
%H Tom 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>, 2015.
%H Tom Copeland, <a href="https://tcjpn.wordpress.com/2008/06/12/mathemagical-forests/">Addendum to Mathemagical Forests</a>, 2010.
%H Tom Copeland, <a href="https://tcjpn.wordpress.com/2008/06/12/mathemagical-forests/">Mathemagical Forests</a>, 2008.
%H A. Dzhumadildaev and D. Yeliussizov, <a href="http://arxiv.org/abs/1408.6764v1">Path decompositions of digraphs and their applications to Weyl algebra</a>, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
%H Askar Dzhumadil'daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
%H Milan 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 D. E. Knuth, <a href="https://arxiv.org/abs/math/9207221">Convolution polynomials</a>, arXiv:math/9207221 [math.CA]; Mathematica J. 2.1 (1992), no. 4, 67-78.
%H Wolfdieter 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 Wolfdieter Lang, <a href="/A035342/a035342.txt">First 10 rows</a>.
%H 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], 2012.
%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
%H E. Neuwirth, <a href="http://homepage.univie.ac.at/erich.neuwirth/papers/TechRep99-05.pdf">Recursively defined combinatorial functions: Extending Galton's board,</a> Discrete Math. 239 (2001) 33-51.
%H Mathias Pétréolle and 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 a(n, m) = Sum_{j=m..n} |A039683(n, j)|*S2(j, m) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to _Wolfdieter Lang_ by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the comment on products of Jabotinsky matrices.
%F a(n, m) = n!*A035324(n, m)/(m!*2^(n-m)), n >= m >= 1; a(n+1, m)= (2*n+m)*a(n, m)+a(n, m-1); a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1.
%F E.g.f. of m-th column: ((x*c(x/2)/sqrt(1-2*x))^m)/m!, where c(x) = g.f. for Catalan numbers A000108.
%F From _Vladimir Kruchinin_, Mar 30 2011: (Start)
%F G.f. (1/sqrt(1-2*x) - 1)^k = Sum_{n>=k} (k!/n!)*a(n,k)*x^n.
%F a(n,k) = 2^(n+k) * n! / (4^n*n*k!) * Sum_{j=0..n-k} (j+k) * 2^(j) * binomial(j+k-1,k-1) * binomial(2*n-j-k-1,n-1). (End)
%F From _Peter Bala_, Nov 25 2011: (Start)
%F E.g.f.: G(x,t) = exp(t*A(x)) = 1 + t*x + (3*t + t^2)*x^2/2! + (15*t + 9*t^2 + t^3)*x^3/3! + ..., where A(x) = -1 + 1/sqrt(1-2*x) satisfies the autonomous differential equation A'(x) = (1+A(x))^3.
%F The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-2*x)*dG/dx, from which follows the recurrence given above.
%F The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^3*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035469 (D = (1+x)^4*d/dx) and A049029 (D = (1+x)^5*d/dx). (End)
%F The n-th row polynomial R(n,x) is given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k>=1} k*(k+2)*...*(k+2*n-2)*x^k/k!. - _Peter Bala_, Jun 22 2014
%F T(n,k) = 2^(k-n)*hypergeom([k-n,k+1],[k-2*n+1],2)*Gamma(2*n-k)/(Gamma(k)*Gamma(n-k+1)). - _Peter Luschny_, Mar 31 2015
%F T(n,k) = 2^n*Sum_{j=1..k} ((-1)^(k-j)*binomial(k, j)*Pochhammer(j/2, n)) / k!. - _Peter Luschny_, Mar 04 2024
%e Matrix begins:
%e 1;
%e 3, 1;
%e 15, 9, 1;
%e 105, 87, 18, 1;
%e 945, 975, 285, 30, 1;
%e ...
%e Combinatoric meaning of a(3,2)=9: The nine increasing path sequences for the three rooted ordered trees with leaves labeled with 1,2,3 and the root labels 0 are: {(0,3),[(0,1),(0,2)]}; {(0,3),[(0,2),(0,1)]}; {(0,3),(0,1,2)}; {(0,1),[(0,3),(0,2)]}; [(0,1),[(0,2),(0,3)]]; [(0,2),[(0,1),(0,3)]]; {(0,2),[(0,3),(0,1)]}; {(0,1),(0,2,3)}; {(0,2),(0,1,3)}.
%p T := (n,k) -> 2^(k-n)*hypergeom([k-n,k+1],[k-2*n+1],2)*GAMMA(2*n-k)/
%p (GAMMA(k)*GAMMA(n-k+1)); for n from 1 to 9 do seq(simplify(T(n,k)),k=1..n) od; # _Peter Luschny_, Mar 31 2015
%p T := (n, k) -> local j; 2^n*add((-1)^(k-j)*binomial(k, j)*pochhammer(j/2, n), j = 1..k)/k!: for n from 1 to 6 do seq(T(n, k), k=1..n) od; # _Peter Luschny_, Mar 04 2024
%t a[n_, k_] := 2^(n+k)*n!/(4^n*n*k!)*Sum[(j+k)*2^(j)*Binomial[j + k - 1, k-1]*Binomial[2*n - j - k - 1, n-1], {j, 0, n-k}]; Flatten[Table[a[n,k], {n, 1, 9}, {k, 1, n}] ] [[1 ;; 40]] (* _Jean-François Alcover_, Jun 01 2011, after _Vladimir Kruchinin_ *)
%o (Maxima) a(n,k):=2^(n+k)*n!/(4^n*n*k!)*sum((j+k)*2^(j)*binomial(j+k-1,k-1)*binomial(2*n-j-k-1,n-1),j,0,n-k) /* _Vladimir Kruchinin_, Mar 30 2011 */
%o (Haskell)
%o a035342 n k = a035342_tabl !! (n-1) !! (k-1)
%o a035342_row n = a035342_tabl !! (n-1)
%o a035342_tabl = map fst $ iterate (\(xs, i) -> (zipWith (+)
%o ([0] ++ xs) $ zipWith (*) [i..] (xs ++ [0]), i + 2)) ([1], 3)
%o -- _Reinhard Zumkeller_, Mar 12 2014
%o (Sage) # uses[bell_matrix from A264428]
%o # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o print(bell_matrix(lambda n: A001147(n+1), 9)) # _Peter Luschny_, Jan 19 2016
%Y The column sequences are A001147, A035101, A035119, ...
%Y Row sums: A049118(n), n >= 1.
%Y Cf. A000108, A035324, A008277, A008297, A094638.
%K easy,nice,nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
%E Simpler name from _Peter Luschny_, Mar 31 2015
|
