|
| |
|
|
A111849
|
|
Column 0 of the matrix logarithm (A111848) of triangle A111845, which shifts columns left and up under matrix 4-th power; these terms are the result of multiplying the element in row n by n!.
|
|
3
| |
|
|
0, 1, 4, 56, 1728, -45696, -159401472, 387212983296, 14722642769657856, -783395638188945997824, -571756408840959817330851840, 603349161280921866200339538247680, 8390141848229920894318007084122311229440
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Let q=4; the g.f. of column k of A111845^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).
|
|
|
FORMULA
| E.g.f. satisfies: x = Sum_{n>=1} -(-1)^n/n!*Prod_{j=0..n-1} A(4^j*x).
|
|
|
EXAMPLE
| E.g.f. A(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4
- 45696/5!*x^5 - 159401472/6!*x^6 +...
where A(x) satisfies:
x = A(x) - A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3!
- A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...
also:
Let G(x) be the g.f. of A111846 (column 0 of A111845), then
G(x) = 1 + x + 4*x^2 + 40*x^3 + 1040*x^4 + 78240*x^5 +...
= 1 + A(x) + A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3!
+ A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! +...
|
|
|
PROG
| (PARI) {a(n, q=4)=local(A=Mat(1), B); if(n<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, if(j==1, B[i, j]=(A^q)[i-1, 1], B[i, j]=(A^q)[i-1, j-1])); )); A=B); B=sum(i=1, #A, -(A^0-A)^i/i); return(n!*B[n+1, 1]))}
|
|
|
CROSSREFS
| Cf. A111848 (matrix log), A111845 (triangle), A111846, A111821 (variant), A111942 (q=-1), A111811 (q=2), A111844 (q=3).
Sequence in context: A113583 A174489 A009563 * A009159 A013055 A012983
Adjacent sequences: A111846 A111847 A111848 * A111850 A111851 A111852
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2005
|
| |
|
|
