A111840
Triangle P, read by rows, that satisfies [P^3](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(3*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.
Original entry on oeis.org
1, 1, 1, 3, 3, 1, 18, 18, 9, 1, 216, 216, 135, 27, 1, 5589, 5589, 4050, 1134, 81, 1, 336555, 336555, 269730, 95256, 9963, 243, 1, 49768101, 49768101, 42724503, 17926839, 2450898, 88938, 729, 1, 18707873562, 18707873562, 16835895603, 8074043145
Offset: 0
Let q=3; 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 + 3/2!*x^2 + 27/3!*x^3 + 486/4!*x^4 + ... (A111844).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(3*x) + m^3/3!*L(x)*L(3*x)*L(3^2*x) +
m^4/4!*L(x)*L(3*x)*L(3^2*x)*L(3^3*x) + ...
Triangle P begins:
1;
1,1;
3,3,1;
18,18,9,1;
216,216,135,27,1;
5589,5589,4050,1134,81,1;
336555,336555,269730,95256,9963,243,1; ...
where P^3 shifts columns left and up one place:
1;
3,1;
18,9,1;
216,135,27,1;
5589,4050,1134,81,1; ...
-
{P(n,k,q=3) = 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(""))
A111848
Matrix log of triangle A111845, which shifts columns left and up under matrix 4th power; 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, 4, 4, 0, 56, 16, 16, 0, 1728, 224, 64, 64, 0, -45696, 6912, 896, 256, 256, 0, -159401472, -182784, 27648, 3584, 1024, 1024, 0, 387212983296, -637605888, -731136, 110592, 14336, 4096, 4096, 0, 14722642769657856, 1548851933184, -2550423552, -2924544, 442368, 57344, 16384, 16384, 0
Offset: 0
Matrix log of A111845, with factorial denominators, begins:
0;
1/1!, 0;
4/2!, 4/1!, 0;
56/3!, 16/2!, 16/1!, 0;
1728/4!, 224/3!, 64/2!, 64/1!, 0;
-45696/5!, 6912/4!, 896/3!, 256/2!, 256/1!, 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
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;
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
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! +...
-
{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]))}
Showing 1-4 of 4 results.
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