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A038455 Triangle read by rows: T(n, k) = [x^k] x*Pochhammer(n + x, n)/(n + x).

%I A038455 #45 Aug 08 2024 06:28:49
%S A038455 1,3,1,20,9,1,210,107,18,1,3024,1650,335,30,1,55440,31594,7155,805,45,
%T A038455 1,1235520,725592,176554,22785,1645,63,1,32432400,19471500,4985316,
%U A038455 705649,59640,3010,84,1,980179200,598482000,159168428,24083892,2267769,136080,5082,108,1
%N A038455 Triangle read by rows: T(n, k) = [x^k] x*Pochhammer(n + x, n)/(n + x).
%C A038455 Original name: A Jabotinsky-triangle related to A006963.
%C A038455 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 A038455 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 A038455 iii) Explicit formula: see Knuth's paper for f(n, m) formula with f(k) = A006963(k + 1).
%C A038455 Bell polynomial of second kind for log(A000108(x)). - _Vladimir Kruchinin_, Mar 26 2013
%C A038455 Also the Bell transform of A006963(n+2). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016
%H A038455 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 A038455 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 A038455 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 A038455 a(n, 1) = A006963(n + 1) = (2*n - 1)!/n!, n >= 1;
%F A038455 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 A038455 E.g.f.: ((1 - sqrt(1 - 4*x))/(2*x))^y. - _Vladeta Jovovic_, May 02 2003
%F A038455 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 A038455 Triangle starts:
%e A038455   [1]       1;
%e A038455   [2]       3,      1;
%e A038455   [3]      20,      9,      1;
%e A038455   [4]     210,    107,     18,     1;
%e A038455   [5]    3024,   1650,    335,    30,    1;
%e A038455   [6]   55440,  31594,   7155,   805,   45,  1;
%e A038455   [7] 1235520, 725592, 176554, 22785, 1645, 63, 1;
%p A038455 # The function BellMatrix is defined in A264428.
%p A038455 # Adds (1,0,0,0, ..) as column 0.
%p A038455 BellMatrix(n -> (2*n+1)!/(n+1)!, 9); # _Peter Luschny_, Jan 28 2016
%p A038455 gf := n -> x*pochhammer(n + x, n)/(n + x):
%p A038455 ser := n -> series(gf(n), x, n + 2):
%p A038455 seq(seq(coeff(ser(n), x, k), k = 1..n), n = 1..9);  # _Peter Luschny_, Jun 27 2024
%t A038455 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A038455 rows = 11; M = BellMatrix[(2#+1)!/(#+1)!&, rows];
%t A038455 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten
%t A038455 (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%o A038455 (Maxima)
%o A038455 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 A038455 Cf. A006963, A000108, A001761, A039619, A039646.
%K A038455 nonn,tabl
%O A038455 1,2
%A A038455 _Wolfdieter Lang_
%E A038455 New name by _Peter Luschny_, Jun 27 2024