A039692 Jabotinsky-triangle related to A039647.

1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 264, 450, 215, 30, 1, 2160, 4114, 2475, 565, 45, 1, 20880, 43512, 30814, 9345, 1225, 63, 1, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1, 3064320, 7235568, 6316316, 2673972, 594489, 69552, 4074, 108, 1

Offset: 1

Author

Wolfdieter Lang

Keywords

• nonn
• tabl

Comments

Triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z)= A(z)/z = 1/(1-z-z^2) where A(z) is the g.f. of the Fibonacci numbers A000045. (Notation of F(z) as in Knuth's paper.)

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)).

E.g.f. for E(n,x): (1 - z - z^2)^(-x).

Explicit a(n,m) formula: see Knuth's paper for f(n,m) formula with f(k)= A039647(n).

E.g.f. for the m-th column sequence: ((-log(1 - z - z^2))^m)/m!.

Also the Bell transform of n!*(F(n)+F(n+2)), F(n) the Fibonacci numbers. For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

Examples

			1;
3, 1;
8, 9, 1;
42, 59, 18, 1;
264, 450, 215, 30, 1;
		

Links

• Vincenzo Librandi, Rows n = 1..50, flattened
• D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.
• Peter Luschny, The Bell transform

Crossrefs

Cf. A039647, A000032, A000045. Another version of this triangle is in A194938.

Programs

Maple

A000032 := proc(n) option remember; coeftayl( (2-x)/(1-x-x^2),x=0,n) ; end: A039647 := proc(n) (n-1)!*A000032(n) ; end: A039692 := proc(n,m) option remember ; if m = 1 then A039647(n) ; else add( binomial(n-1,j-1)*A039647(j)*procname(n-j,m-1),j=1..n-m+1) ; fi; end: # R. J. Mathar, Jun 01 2009

Mathematica

t[n_, m_] := n!*Sum[StirlingS1[k, m]*Binomial[k, n-k]*(-1)^(k+m)/k!, {k, m, n}]; Table[t[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013, after Vladimir Kruchinin *)

Maxima

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

PARI

T(n,m) = n!*sum(k=m,n, (stirling(k,m,1)*binomial(k,n-k))*(-1)^(k+m)/k!);
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print());
/* Joerg Arndt, Mar 27 2013 */

Sage

# uses[bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 to the left side of the triangle.
bell_matrix(lambda n: factorial(n)*(fibonacci(n)+fibonacci(n+2)), 8) # Peter Luschny, Jan 16 2016

Formula

a(n, 1)= A039647(n)=(n-1)!*L(n), L(n) := A000032(n) (Lucas); a(n, m) = Sum_{j=1..n-m+1} binomial(n-1, j-1)*A039647(j)*a(n-j, m-1), n >= m >= 2.

Conjectured row sums: sum_{m=1..n} a(n,m) = A005442(n). - R. J. Mathar, Jun 01 2009

T(n,m) = n! * Sum_{k=m..n} stirling1(k,m)*binomial(k,n-k)*(-1)^(k+m)/k!. - Vladimir Kruchinin, Mar 26 2013