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Triangle read by rows: T(n, k) = [x^k] x*Pochhammer(n + x, n)/(n + x).
5

%I #45 Aug 08 2024 06:28:49

%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 705649,59640,3010,84,1,980179200,598482000,159168428,24083892,2267769,136080,5082,108,1

%N Triangle read by rows: T(n, k) = [x^k] x*Pochhammer(n + x, n)/(n + x).

%C Original name: 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_{m=1..n} a(n, m)*x^m, E(0, x) = 1, are exponential convolution polynomials: E(n, x + y) = Sum_{k=0..n} binomial(n, k)*E(k, x)*E(n-k, y), (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;

%F a(n, m) = Sum_{j=1..n-m+1} binomial(n - 1, j - 1)*A006963(j + 1)*a(n - j, m - 1), n >= m >= 2.

%F E.g.f.: ((1 - sqrt(1 - 4*x))/(2*x))^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

%e Triangle starts:

%e [1] 1;

%e [2] 3, 1;

%e [3] 20, 9, 1;

%e [4] 210, 107, 18, 1;

%e [5] 3024, 1650, 335, 30, 1;

%e [6] 55440, 31594, 7155, 805, 45, 1;

%e [7] 1235520, 725592, 176554, 22785, 1645, 63, 1;

%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

%p gf := n -> x*pochhammer(n + x, n)/(n + x):

%p ser := n -> series(gf(n), x, n + 2):

%p seq(seq(coeff(ser(n), x, k), k = 1..n), n = 1..9); # _Peter Luschny_, Jun 27 2024

%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; M = BellMatrix[(2#+1)!/(#+1)!&, rows];

%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten

%t (* _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_

%E New name by _Peter Luschny_, Jun 27 2024