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A111844
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Column 0 of the matrix logarithm (A111843) of triangle A111840, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.
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4
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0, 1, 3, 27, 486, 7776, -2423196, -97338996, 5883879500784, 548540050402080, -1737375315124971951360, -405928706169160555680960, 60788545124934395018363657569920, 36207408592259278909089966337224960, -237458310218887960183820317532070376189904640
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Let q=3; the g.f. of column k of A111840^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).
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FORMULA
| E.g.f. satisfies: x = -Sum_{n>=1} Prod_{j=0..n-1} -A(3^j*x)/(j+1).
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EXAMPLE
| E.g.f. A(x) = x + 3/2!*x^2 + 27/3!*x^3 + 486/4!*x^4 + 7776/5!*x^5
- 2423196/6!*x^6 - 97338996/7!*x^7 +...
where A(x) satisfies:
x = A(x) - A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3!
- A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
also:
Let G(x) be the g.f. of A111841 (column 0 of A111840), then
G(x) = 1 + x + 3*x^2 + 18*x^3 + 216*x^4 + 5589*x^5 + 336555*x^6 +...
= 1 + A(x) + A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3!
+ A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! +...
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PROG
| (PARI) {a(n, q=3)=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]))}
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CROSSREFS
| Cf. A111843 (matrix log), A111840 (triangle), A111841, A111816 (variant), A111941 (q=-1), A111843 (q=3), A111848 (q=4).
Sequence in context: A159601 A193541 A193544 * A118714 A089506 A062496
Adjacent sequences: A111841 A111842 A111843 * A111845 A111846 A111847
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KEYWORD
| sign
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2005
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