A111844
Column 0 of the matrix logarithm (A111843) of triangle A111840, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.
Original entry on oeis.org
0, 1, 3, 27, 486, 7776, -2423196, -97338996, 5883879500784, 548540050402080, -1737375315124971951360, -405928706169160555680960, 60788545124934395018363657569920, 36207408592259278909089966337224960, -237458310218887960183820317532070376189904640
Offset: 0
E.g.f. A(x) = x + 3/2!*x^2 + 27/3!*x^3 + 486/4!*x^4 + 7776/5!*x^5
- 2423196/6!*x^6 - 97338996/7!*x^7 +...
where A(x) satisfies:
x = A(x) - A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3!
- A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
also:
Let G(x) be the g.f. of A111841 (column 0 of A111840), then
G(x) = 1 + x + 3*x^2 + 18*x^3 + 216*x^4 + 5589*x^5 + 336555*x^6 +...
= 1 + A(x) + A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3!
+ A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! +...
-
{a(n,q=3)=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]))}
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(""))
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!.
Original entry on oeis.org
0, 1, 2, 10, 88, 1096, 11856, -402480, -1891968, 36024603264, 359905478400, -53686393014816000, -644141701131494400, 1790653231402788752593920, 25068910772059830672353280, -1280832036591718248285105113241600
Offset: 0
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! +...
-
{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))}
A111810
Matrix log of triangle A098539, 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, 2, 2, 0, 10, 4, 4, 0, 88, 20, 8, 8, 0, 1096, 176, 40, 16, 16, 0, 11856, 2192, 352, 80, 32, 32, 0, -402480, 23712, 4384, 704, 160, 64, 64, 0, -1891968, -804960, 47424, 8768, 1408, 320, 128, 128, 0, 36024603264, -3783936, -1609920, 94848, 17536, 2816, 640, 256, 256, 0
Offset: 0
Matrix log of A098539, with factorial denominators, begins:
0;
1/1!, 0;
2/2!, 2/1!, 0;
10/3!, 4/2!, 4/1!, 0;
88/4!, 20/3!, 8/2!, 8/1!, 0;
1096/5!, 176/4!, 40/3!, 16/2!, 16/1!, 0;
11856/6!, 2192/5!, 352/4!, 80/3!, 32/2!, 32/1!, 0; ...
Showing 1-4 of 4 results.
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