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Column 0 of the matrix logarithm (A111818) of triangle A078536, which shifts columns left and up under matrix 4-th 4th power; these terms are the result of multiplying the element in row n by n!.

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editing

Gottfried Helms and _Paul D. Hanna (pauldhanna(AT)juno.com), _, Aug 22 2005

_Gottfried Helms (helms(AT)uni-kassel.de) _ and Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2005

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Gottfried Helms (helms(AT)uni-kassel.de) and Paul D . Hanna (pauldhanna(AT)juno.com), Aug 22 2005

0,43

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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!.

0, 1, -2, 2, 840, -76056, -158761104, 390564896784, 14713376473366656, -783793232940393380736, -571732910947761663424746240, 603368029500890443054004423520000, 8390120127886533420753746115877557580800

0,4

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).

E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(4^j*x)/(j+1).

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! + ...

(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))}

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).

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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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