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; ...
A111846
Number of partitions of 4^n - 1 into powers of 4, also equals column 0 of triangle A111845, which shifts columns left and up under matrix 4th power.
Original entry on oeis.org
1, 1, 4, 40, 1040, 78240, 18504256, 14463224448, 38544653734144, 357896006503348736, 11766320092785122862080, 1387031702368547767793690624, 592262859312707222259571097997312
Offset: 0
G.f. A(x) = 1 + L(x) + L(x)*L(4*x)/2! + L(x)*L(4*x)*L(4^2*x)/3!
+ L(x)*L(4*x)*L(4^2*x)*L(4^3*x)/4! + ...
where L(x) satisfies:
x = L(x) - L(x)*L(4*x)/2! + L(x)*L(4*x)*L(4^2*x)/3! +- ...
and L(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4 +....(A111849).
-
{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);return(A[n+1,1]))}
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]))}
A111847
Row sums of triangle A111845, which shifts columns left and up under matrix 4th power.
Original entry on oeis.org
1, 2, 9, 97, 2689, 214017, 53130241, 43283609601, 119521939222529, 1144341237628100609, 38638551719263573098497, 4662529388979590206324834305, 2032489532637330252763496597356545
Offset: 0
-
{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); return(sum(k=0,n,A[n+1,k+1])))}
A111818
Matrix log of triangle A078536, 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, -2, 4, 0, 2, -8, 16, 0, 840, 8, -32, 64, 0, -76056, 3360, 32, -128, 256, 0, -158761104, -304224, 13440, 128, -512, 1024, 0, 390564896784, -635044416, -1216896, 53760, 512, -2048, 4096, 0, 14713376473366656, 1562259587136, -2540177664, -4867584, 215040, 2048, -8192, 16384, 0
Offset: 0
Matrix log of A078536, with factorial denominators, begins:
0;
1/1!, 0;
-2/2!, 4/1!, 0;
2/3!, -8/2!, 16/1!, 0;
840/4!, 8/3!, -32/2!, 64/1!, 0;
-76056/5!, 3360/4!, 32/3!, -128/2!, 256/1!, 0;
-158761104/6!, -304224/5!, 13440/4!, 128/3!, -512/2!, 1024/1!, 0;
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