|
| |
|
|
A039692
|
|
Jabotinsky-triangle related to A039647.
|
|
37
| |
|
|
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
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
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(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)).
Egf 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).
Egf for m-th column sequence: ((-ln(1-z-z^2))^m)/m!.
|
|
|
REFERENCES
| D. E. Knuth, Convolution polynomials, The Mathematica J., 2.1 (1992) 67-78.
|
|
|
FORMULA
| a(n, 1)= A039647(n)=(n-1)!*L(n), L(n) := A000032(n) (Lucas); a(n, m) = sum(binomial(n-1, j-1)*A039647(j)*a(n-j, m-1), j=1..n-m+1), n >= m >= 2.
Conjectured row sums: sum_{m=1..n} a(n,m) = A005442(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 01 2009]
|
|
|
EXAMPLE
| 1;
3, 1;
8, 9, 1;
42, 59, 18, 1;
264, 450, 215, 30, 1;
|
|
|
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: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 01 2009]
|
|
|
CROSSREFS
| Cf. A039647, A000032, A000045. Another version of this triangle is in A194938.
Sequence in context: A176103 A076238 A008298 * A071815 A178301 A120236
Adjacent sequences: A039689 A039690 A039691 * A039693 A039694 A039695
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
|
| |
|
|
