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A111834 Column 0 of the matrix logarithm (A111833) of triangle A111830, which shifts columns left and up under matrix 7th power; these terms are the result of multiplying the element in row n by n!. 8
0, 1, -5, 83, 16110, -40097784, -388036363380, 82804198261002036, 50475967918183333160880, -711988160501968313699728393632, -26438313284970847487368499812182785280, 22571673265500745067336177578868612107537514880 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let q=7; the g.f. of column k of A111830^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

LINKS

Table of n, a(n) for n=0..11.

FORMULA

E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(7^j*x)/(j+1).

EXAMPLE

A(x) = x - 5/2!*x^2 + 83/3!*x^3 + 16110/4!*x^4 - 40097784/5!*x^5 +...

where e.g.f. A(x) satisfies:

x/(1-x) = A(x) + A(x)*A(7*x)/2! + A(x)*A(7*x)*A(7^2*x)/3! +

A(x)*A(7*x)*A(7^2*x)*A(7^3*x)/4! + ...

Let G(x) be the g.f. of A111831 (column 1 of A111830), then

G(x) = 1 + 7*A(x) + 7^2*A(x)*A(7*x)/2! +

7^3*A(x)*A(7*x)*A(7^2*x)/3! +

7^4*A(x)*A(7*x)*A(7^2*x)*A(7^3*x)/4! + ...

PROG

(PARI) {a(n, q=7)=local(A=x/(1-x+x*O(x^n))); for(i=1, n, A=x/(1-x)/(1+sum(j=1, n, prod(k=1, j, subst(A, x, q^k*x))/(j+1)!))); return(n!*polcoeff(A, n))}

CROSSREFS

Cf. A111830 (triangle), A111831, A111833 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111839 (q=8).

Sequence in context: A215172 A301811 A216146 * A271219 A187587 A006471

Adjacent sequences:  A111831 A111832 A111833 * A111835 A111836 A111837

KEYWORD

sign

AUTHOR

Gottfried Helms and Paul D. Hanna, Aug 22 2005

STATUS

approved

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Last modified September 25 19:42 EDT 2020. Contains 337344 sequences. (Running on oeis4.)