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Search: id:A111816
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| A111816 |
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Column 0 of the matrix logarithm (A111815) of triangle A078122, 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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9
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| 0, 1, -1, -3, 150, 1236, -2555748, -64342116, 5885700899760, 442646611978752, -1737387344860364226240, -367706581563500487774720, 60788555325888838346137808787840, 34626906551623392401873575206240000, -237458311254822429335982538087618909465992960
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Let q=3; the g.f. of column k of A078122^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
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E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(3^j*x)/(j+1).
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EXAMPLE
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E.g.f.: A(x) = x - 1/2!*x^2 - 3/3!*x^3 + 150/4!*x^4 + 1236/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-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! + ...
Let G(x) be the g.f. of A078124 (column 1 of A078122), then
G(x) = 1 + 3*A(x) + 3^2*A(x)*A(3*x)/2! +
3^3*A(x)*A(3*x)*A(3^2*x)/3! +
3^4*A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
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PROGRAM
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(PARI) {a(n, q=3)=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))}
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CROSSREFS
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Cf. A078122 (triangle), A078124, A111815 (matrix log); A110505 (q=-1), A111814 (q=2), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).
Sequence in context: A003009 A037122 A118840 this_sequence A157555 A157578 A137802
Adjacent sequences: A111813 A111814 A111815 this_sequence A111817 A111818 A111819
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KEYWORD
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sign
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AUTHOR
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Gottfried Helms (helms(AT)uni-kassel.de) and Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2005
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