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 A111829 Column 0 of the matrix logarithm (A111828) of triangle A111825, which shifts columns left and up under matrix 6th power; these terms are the result of multiplying the element in row n by n!. 8
 0, 1, -4, 42, 7296, -7931976, -45557382240, 3064554175021200, 801993619807364206080, -2618439032548254776387771520, -30580166025709706974876961026475520, 4440597519115996836838709580481861376121600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Let q=6; the g.f. of column k of A111825^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x). LINKS FORMULA E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(6^j*x)/(j+1). EXAMPLE A(x) = x - 4/2!*x^2 + 42/3!*x^3 + 7296/4!*x^4 - 7931976/5!*x^5 +... where e.g.f. A(x) satisfies: x/(1-x) = A(x) + A(x)*A(6*x)/2! + A(x)*A(6*x)*A(6^2*x)/3! + A(x)*A(6*x)*A(6^2*x)*A(6^3*x)/4! + ... Let G(x) be the g.f. of A111826 (column 1 of A111825), then G(x) = 1 + 6*A(x) + 6^2*A(x)*A(6*x)/2! + 6^3*A(x)*A(6*x)*A(6^2*x)/3! + 6^4*A(x)*A(6*x)*A(6^2*x)*A(6^3*x)/4! + ... PROG (PARI) {a(n, q=6)=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. A111825 (triangle), A111826, A111828 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111834 (q=7), A111839 (q=8). Sequence in context: A220180 A134356 A156479 * A200000 A198209 A220774 Adjacent sequences:  A111826 A111827 A111828 * A111830 A111831 A111832 KEYWORD sign AUTHOR Gottfried Helms and Paul D. Hanna, Aug 22 2005 STATUS approved

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Last modified January 21 11:50 EST 2019. Contains 319354 sequences. (Running on oeis4.)