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A038455 A Jabotinsky-triangle related to A006963. 5
1, 3, 1, 20, 9, 1, 210, 107, 18, 1, 3024, 1650, 335, 30, 1, 55440, 31594, 7155, 805, 45, 1, 1235520, 725592, 176554, 22785, 1645, 63, 1, 32432400, 19471500, 4985316, 705649, 59640, 3010, 84, 1, 980179200, 598482000, 159168428, 24083892, 2267769 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

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).)

iii) Explicit formula: see Knuth's paper for f(n,m) formula with f(k)= A006963(k+1).

Bell polynomial of second kind for log(A000108(x). - Vladimir Kruchinin, Mar 26 2013

Also the Bell transform of A006963(n+2). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

LINKS

Table of n, a(n) for n=1..41.

D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.

J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.

FORMULA

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.

E.g.f.: ((1-sqrt(1-4*x))/x/2)^y. - Vladeta Jovovic, May 02 2003

a(n,m) = (n-1)!*(sum_{k=m..n} stirling1(k,m)*binomial(2*n,n-k)/(k-1)!). - Vladimir Kruchinin, Mar 26 2013

MAPLE

# The function BellMatrix is defined in A264428.

# Adds (1, 0, 0, 0, ..) as column 0.

BellMatrix(n -> (2*n+1)!/(n+1)!, 9); # Peter Luschny, Jan 28 2016

MATHEMATICA

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

rows = 11;

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

Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 24 2018, after Peter Luschny *)

PROG

(Maxima)

a(n, m):=(n-1)!*(sum((stirling1(k, m)*binomial(2*n, n-k))/(k-1)!, k, m, n)); /* Vladimir Kruchinin, Mar 26 2013 */

CROSSREFS

Cf. A006963, A000108, A001761, A039619, A039646.

Sequence in context: A192721 A002380 A274075 * A067802 A181832 A139723

Adjacent sequences:  A038452 A038453 A038454 * A038456 A038457 A038458

KEYWORD

nonn,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified October 17 12:50 EDT 2018. Contains 316280 sequences. (Running on oeis4.)