A111846 -id:A111846 - cp's OEIS frontend

A111845 Triangle P, read by rows, that satisfies [P^4](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(4*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(k,k)=1 and P(k+1,1)=P(k+1,0) for k>=0.

1, 1, 1, 4, 4, 1, 40, 40, 16, 1, 1040, 1040, 544, 64, 1, 78240, 78240, 48960, 8320, 256, 1, 18504256, 18504256, 13110400, 2878720, 131584, 1024, 1, 14463224448, 14463224448, 11192599808, 2982187520, 180270080, 2099200, 4096, 1

Offset: 0

Paul D. Hanna, Aug 23 2005

Column 0 and column 1 are equal for n>0.

			Let q=4; the g.f. of column k of matrix power P^m is:
1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +
(m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +
(m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...
where L(x) satisfies:
x = L(x) - L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! +- ...
and L(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4 + ... (A111849).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(4*x) + m^3/3!*L(x)*L(4*x)*L(4^2*x) +
m^4/4!*L(x)*L(4*x)*L(4^2*x)*L(4^3*x) + ...
Triangle P begins:
1;
1,1;
4,4,1;
40,40,16,1;
1040,1040,544,64,1;
78240,78240,48960,8320,256,1;
18504256,18504256,13110400,2878720,131584,1024,1; ...
where P^4 shifts columns left and up one place:
1;
4,1;
40,16,1;
1040,544,64,1;
78240,48960,8320,256,1; ...

Cf. A111846 (column 0), A111847 (row sums), A111848 (matrix log), A111840 (q=3), A078536 (variant).

{P(n,k,q=4) = my(A=Mat(1),B);if(nPaul D. Hanna, Jul 11 2025):
for(n=0,10, for(k=0,n, print1(P(n,k),", ")); print(""))

Let q=4; the g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} L(q^j*x) where L(x) satisfies: x = Sum_{n>=1} -(-1)^n/n!*Product_{j=0..n-1} L(q^j*x); L(x) equals the g.f. of column 0 of the matrix log of P (A111849).

A111849 Column 0 of the matrix logarithm (A111848) of triangle A111845, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, 4, 56, 1728, -45696, -159401472, 387212983296, 14722642769657856, -783395638188945997824, -571756408840959817330851840, 603349161280921866200339538247680, 8390141848229920894318007084122311229440

Offset: 0

Paul D. Hanna, Aug 23 2005

sign

Let q=4; the g.f. of column k of A111845^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. A(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4
- 45696/5!*x^5 - 159401472/6!*x^6 +...
where A(x) satisfies:
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! + ...
also:
Let G(x) be the g.f. of A111846 (column 0 of A111845), then
G(x) = 1 + x + 4*x^2 + 40*x^3 + 1040*x^4 + 78240*x^5 +...
= 1 + 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! +...

Cf. A111848 (matrix log), A111845 (triangle), A111846, A111821 (variant), A111942 (q=-1), A111811 (q=2), A111844 (q=3).

{a(n,q=4)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,if(j==1,B[i,j]=(A^q)[i-1,1], B[i,j]=(A^q)[i-1,j-1]));));A=B); B=sum(i=1,#A,-(A^0-A)^i/i);return(n!*B[n+1,1]))}

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

cp's OEIS Frontend

A111845 Triangle P, read by rows, that satisfies [P^4](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(4*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(k,k)=1 and P(k+1,1)=P(k+1,0) for k>=0.

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