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%I #33 Aug 12 2022 15:12:59
%S 1,3,1,20,9,1,210,107,18,1,3024,1650,335,30,1,55440,31594,7155,805,45,
%T 1,1235520,725592,176554,22785,1645,63,1,32432400,19471500,4985316,
%U 705649,59640,3010,84,1,980179200,598482000,159168428,24083892,2267769
%N A Jabotinsky-triangle related to A006963.
%C i) This triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z)= c(z) with c(z) the g.f. for the Catalan numbers A000108. (Notation of F(z) as in Knuth's paper).
%C ii) E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x)=1, are exponential convolution polynomials: E(n,x+y) = sum(binomial(n,k)*E(k,x)*E(n-k,y),k=0..n) (cf. Knuth's paper with E(n,x)= n!*F(n,x).)
%C iii) Explicit formula: see Knuth's paper for f(n,m) formula with f(k)= A006963(k+1).
%C Bell polynomial of second kind for log(A000108(x). - _Vladimir Kruchinin_, Mar 26 2013
%C Also the Bell transform of A006963(n+2). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016
%H Priyavrat Deshpande and Krishna Menon, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2022/23.pdf">A statistic for regions of braid deformations</a>, Séminaire Lotharingien de Combinatoire (2022) Vol. 86, Issue B, Art. No. 23.
%H D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, arXiv:math/9207221 [math.CA], 1992; Mathematica J. 2.1 (1992), no. 4, 67-78.
%H J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0512570">Noncommutative Symmetric Functions and Lagrange Inversion</a>, arXiv:math/0512570 [math.CO], 2005-2006.
%F a(n, 1) = A006963(n+1)=(2*n-1)!/n!, n >= 1; a(n, m) = sum(binomial(n-1, j-1)*A006963(j+1)*a(n-j, m-1), j=1..n-m+1), n >= m >= 2.
%F E.g.f.: ((1-sqrt(1-4*x))/x/2)^y. - _Vladeta Jovovic_, May 02 2003
%F a(n,m) = (n-1)!*(sum_{k=m..n} stirling1(k,m)*binomial(2*n,n-k)/(k-1)!). - _Vladimir Kruchinin_, Mar 26 2013
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> (2*n+1)!/(n+1)!, 9); # _Peter Luschny_, Jan 28 2016
%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t rows = 11;
%t M = BellMatrix[(2#+1)!/(#+1)!&, rows];
%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%o (Maxima)
%o a(n,m):=(n-1)!*(sum((stirling1(k,m)*binomial(2*n,n-k))/(k-1)!,k,m,n)); /* _Vladimir Kruchinin_, Mar 26 2013 */
%Y Cf. A006963, A000108, A001761, A039619, A039646.
%K nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
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