A111814 Column 0 of the matrix logarithm (A111813) of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

0, 1, 0, -2, 0, 216, 0, -568464, 0, 36058658688, 0, -53694310935340800, 0, 1790669979087018171448320, 0, -1280832788659041410080025283840000, 0, 18961468161294510864200732026858464699187200, 0

Offset

0,4

Comments

Surprisingly, the e.g.f. A(x) is an odd function: A(-x) = -A(x). Let q=2; the g.f. of column k of A078121^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

Links

Paul D. Hanna, Table of n, a(n), n== 0..50.

Formula

E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(2^j*x)/(j+1). E.g.f. satisfies: x/(1-x^2) = Sum_{n>=1}Prod_{j=0..2*n}A(2^j*x)/(j+1).

Example

E.g.f.: A(x) = x - 2/3!*x^3 + 216/5!*x^5 - 568464/7!*x^7 + ...
where A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(2*x)/2! + A(x)*A(2*x)*A(2^2*x)/3!
+ A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! + ...
also:
x/(1-x^2) = A(x) + A(x)*A(2*x)*A(2^2*x)/3!
+ A(x)*A(2*x)*A(2^2*x)*A(2^3*x)*A(2^4*x)/5! + ...
Let G(x) be the g.f. of A002577 (column 1 of A078121), then
G(x) = 1 + 2*A(x) + 2^2*A(x)*A(2*x)/2! +
2^3*A(x)*A(2*x)*A(2^2*x)/3! +
2^4*A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! + ...

Prog

(PARI) {a(n, q=2)=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. A078121 (triangle), A002577, A111813 (matrix log); A110505 (q=-1), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

Sequence in context: A012335 A012331 A037096 * A036938 A344463 A221750

Adjacent sequences: A111811 A111812 A111813 * A111815 A111816 A111817

Keyword

sign

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Status

approved