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A111819
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Column 0 of the matrix logarithm (A111818) of triangle A078536, 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!.
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7
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0, 1, -2, 2, 840, -76056, -158761104, 390564896784, 14713376473366656, -783793232940393380736, -571732910947761663424746240, 603368029500890443054004423520000, 8390120127886533420753746115877557580800
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OFFSET
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0,3
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COMMENTS
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Let q=4; the g.f. of column k of A078536^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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LINKS
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Table of n, a(n) for n=0..12.
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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(4^j*x)/(j+1).
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EXAMPLE
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A(x) = x - 2/2!*x^2 + 2/3!*x^3 + 840/4!*x^4 - 76056/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-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! + ...
Let G(x) be the g.f. of A111817 (column 1 of A078536), then
G(x) = 1 + 4*A(x) + 4^2*A(x)*A(4*x)/2! +
4^3*A(x)*A(4*x)*A(4^2*x)/3! +
4^4*A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...
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PROG
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(PARI) {a(n, q=4)=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. A078536 (triangle), A111817, A111818 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).
Sequence in context: A013556 A093596 A095304 * A079237 A013510 A013504
Adjacent sequences: A111816 A111817 A111818 * A111820 A111821 A111822
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KEYWORD
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sign
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AUTHOR
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Gottfried Helms and Paul D. Hanna, Aug 22 2005
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STATUS
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approved
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