A111810 -id:A111810 - cp's OEIS frontend

A111811 Column 0 of the matrix logarithm (A111810) of triangle A098539, 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, 2, 10, 88, 1096, 11856, -402480, -1891968, 36024603264, 359905478400, -53686393014816000, -644141701131494400, 1790653231402788752593920, 25068910772059830672353280, -1280832036591718248285105113241600

Offset: 0

Paul D. Hanna, Aug 22 2005

sign

Let q=2; the g.f. of column k of A098539^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

			A(x) = x + 2/2!*x^2 + 10/3!*x^3 + 88/4!*x^4 + 1096/5!*x^5 +...
where e.g.f. A(x) satisfies:
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) = A(x) - 2*A(x)*A(2*x)/2! + 2^2*A(x)*A(2*x)*A(2^2*x)/3! - 2^3*A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! +...
Let G(x) be the g.f. of A002449 (column 1 of A098539), then
(G(x)-1)/x = 1 + 2*x + 6*x^2 + 26*x^3 + 166*x^4 + 1626*x^5 +...
= 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! +...

Cf. A111810 (matrix log), A098539 (triangle), A002449, A111814 (variant), A111941 (q=-1), A111843 (q=3), A111848 (q=4).

{a(n,q=2)=local(A=x+x*O(x^n));for(i=1,n, A=x/(1+sum(j=1,n,prod(k=1,j,-subst(A,x,q^k*x))/(j+1)!))); return(n!*polcoeff(A,n))}

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

A111813 Matrix log of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, -2, 0, 4, 0, 0, -4, 0, 8, 0, 216, 0, -8, 0, 16, 0, 0, 432, 0, -16, 0, 32, 0, -568464, 0, 864, 0, -32, 0, 64, 0, 0, -1136928, 0, 1728, 0, -64, 0, 128, 0, 36058658688, 0, -2273856, 0, 3456, 0, -128, 0, 256, 0, 0, 72117317376, 0, -4547712, 0, 6912, 0, -256, 0, 512, 0

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 2^k multiplied by column 0 (A111814) when ignoring zeros above the diagonal.

			Matrix log of A078121, with factorial denominators, begins:
0;
1/1!, 0;
0/2!, 2/1!, 0;
-2/3!, 0/2!, 4/1!, 0;
0/4!, -4/3!, 0/2!, 8/1!, 0;
216/5!, 0/4!, -8/3!, 0/2!, 16/1!, 0;
0/6!, 432/5!, 0/4!, -16/3!, 0/2!, 32/1!, 0;
-568464/7!, 0/6!, 864/5!, 0/4!, -32/3!, 0/2!, 64/1!, 0; ...

Cf. A078121, A111814 (column 0), A111810 (variant); log matrices: A110504 (q=-1), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

PARI
```
T(n,k,q=2)=local(A=Mat(1),B);if(n
				
```

T(n, k) = 2^k*T(n-k, 0) = A111814(n-k) for n>=k>=0.

A111843 Matrix log of triangle A111840, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, 3, 3, 0, 27, 9, 9, 0, 486, 81, 27, 27, 0, 7776, 1458, 243, 81, 81, 0, -2423196, 23328, 4374, 729, 243, 243, 0, -97338996, -7269588, 69984, 13122, 2187, 729, 729, 0, 5883879500784, -292016988, -21808764, 209952, 39366, 6561, 2187, 2187, 0

Offset: 0

Paul D. Hanna, Aug 23 2005

Column k equals 3^k multiplied by column 0 (A111844) when ignoring zeros above the diagonal.

			Matrix log of A111840, with factorial denominators, begins:
0;
1/1!, 0;
3/2!, 3/1!, 0;
27/3!, 9/2!, 9/1!, 0;
486/4!, 81/3!, 27/2!, 27/1!, 0;
7776/5!, 1458/4!, 243/3!, 81/2!, 81/1!, 0;
-2423196/6!, 23328/5!, 4374/4!, 729/3!, 243/2!, 243/1!, 0;

Cf. A111840 (triangle), A111844 (column 0), A111815 (variant), A111941 (q=-1), A111810 (q=2), A111848 (q=4).

PARI
```
T(n,k,q=3)=local(A=Mat(1),B);if(n
				
```

T(n, k) = 3^k*T(n-k, 0) = 3^k*A111844(n-k) for n>=k>=0.

cp's OEIS Frontend

A111811 Column 0 of the matrix logarithm (A111810) of triangle A098539, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

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A111813 Matrix log of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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PARI

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A111843 Matrix log of triangle A111840, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula