A111838 -id:A111838 - cp's OEIS frontend

A111839 Column 0 of the matrix logarithm (A111838) of triangle A111835, which shifts columns left and up under matrix 8th power; these terms are the result of multiplying the element in row n by n!.

0, 1, -6, 142, 31800, -159468264, -2481298801008, 1414130111428687344, 1827317023092830201950080, -89946874545119714361987192509568, -9262235489215916508714844705185660161280

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Let q=8; the g.f. of column k of A111825^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 - 6/2!*x^2 + 142/3!*x^3 + 31800/4!*x^4 - 159468264/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(8*x)/2! + A(x)*A(8*x)*A(8^2*x)/3! +
A(x)*A(8*x)*A(8^2*x)*A(8^3*x)/4! + ...
Let G(x) be the g.f. of A111836 (column 1 of A111835), then
G(x) = 1 + 8*A(x) + 8^2*A(x)*A(8*x)/2! +
8^3*A(x)*A(8*x)*A(8^2*x)/3! +
8^4*A(x)*A(8*x)*A(8^2*x)*A(8^3*x)/4! + ...

Cf. A111835 (triangle), A111836, A111838 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7).

{a(n,q=8)=local(A=x/(1-x+x*O(x^n)));for(i=1,n, A=x/(1-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/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(8^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.

A111815 Matrix log of triangle A078122, 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, -1, 3, 0, -3, -3, 9, 0, 150, -9, -9, 27, 0, 1236, 450, -27, -27, 81, 0, -2555748, 3708, 1350, -81, -81, 243, 0, -64342116, -7667244, 11124, 4050, -243, -243, 729, 0, 5885700899760, -193026348, -23001732, 33372, 12150, -729, -729, 2187, 0

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 3^k multiplied by column 0 (A111816) when ignoring zeros above the diagonal.

			Matrix log of A078122, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, 3/1!, 0;
-3/3!, -3/2!, 9/1!, 0;
150/4!, -9/3!, -9/2!, 27/1!, 0;
1236/5!, 450/4!, -27/3!, -27/2!, 81/1!, 0;
-2555748/6!, 3708/5!, 1350/4!, -81/3!, -81/2!, 243/1!, 0; ...

Cf. A078122, A111816 (column 0), A111840 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

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

T(n, k) = 3^k*T(n-k, 0) = A111816(n-k) for n>=k>=0.

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

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 4^k multiplied by column 0 (A111819) when ignoring zeros above the diagonal.

			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;

Cf. A078536, A111819 (column 0), A111845 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

PARI
```
T(n,k,q=4)=local(A=Mat(1),B);if(n
				
```

T(n, k) = 4^k*T(n-k, 0) = A111819(n-k) for n>=k>=0.

A111823 Matrix log of triangle A111820, which shifts columns left and up under matrix 5th 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, -3, 5, 0, 16, -15, 25, 0, 2814, 80, -75, 125, 0, -1092180, 14070, 400, -375, 625, 0, -3603928080, -5460900, 70350, 2000, -1875, 3125, 0, 58978973128440, -18019640400, -27304500, 351750, 10000, -9375, 15625, 0, 5974833380453777520

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 5^k multiplied by column 0 (A111824) when ignoring zeros above the diagonal.

			Matrix log of A111820, with factorial denominators, begins:
0;
1/1!, 0;
-3/2!, 5/1!, 0;
16/3!, -15/2!, 25/1!, 0;
2814/4!, 80/3!, -75/2!, 125/1!, 0;
-1092180/5!, 14070/4!, 400/3!, -375/2!, 625/1!, 0; ...

Cf. A111820, A111824 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111828 (q=6), A111833 (q=7), A111838 (q=8).

PARI
```
T(n,k,q=5)=local(A=Mat(1),B);if(n
				
```

T(n, k) = 5^k*T(n-k, 0) = A111824(n-k) for n>=k>=0.

A111828 Matrix log of triangle A111825, which shifts columns left and up under matrix 6th 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, 6, 0, 42, -24, 36, 0, 7296, 252, -144, 216, 0, -7931976, 43776, 1512, -864, 1296, 0, -45557382240, -47591856, 262656, 9072, -5184, 7776, 0, 3064554175021200, -273344293440, -285551136, 1575936, 54432, -31104, 46656, 0, 801993619807364206080

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 6^k multiplied by column 0 (A111829) when ignoring zeros above the diagonal.

			Matrix log of A111825, with factorial denominators, begins:
0;
1/1!, 0;
-4/2!, 6/1!, 0;
42/3!, -24/2!, 36/1!, 0;
7296/4!, 252/3!, -144/2!, 216/1!, 0;
-7931976/5!, 43776/4!, 1512/3!, -864/2!, 1296/1!, 0; ...

Cf. A111825, A111829 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111833 (q=7), A111838 (q=8).

PARI
```
T(n,k,q=6)=local(A=Mat(1),B);if(n
				
```

T(n, k) = 6^k*T(n-k, 0) = A111829(n-k) for n>=k>=0.

A111833 Matrix log of triangle A111830, which shifts columns left and up under matrix 7th 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, -5, 7, 0, 83, -35, 49, 0, 16110, 581, -245, 343, 0, -40097784, 112770, 4067, -1715, 2401, 0, -388036363380, -280684488, 789390, 28469, -12005, 16807, 0, 82804198261002036, -2716254543660, -1964791416, 5525730, 199283, -84035, 117649, 0

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 7^k multiplied by column 0 (A111834) when ignoring zeros above the diagonal.

			Matrix log of A111830, with factorial denominators, begins:
0;
1/1!, 0;
-5/2!, 7/1!, 0;
83/3!, -35/2!, 49/1!, 0;
16110/4!, 581/3!, -245/2!, 343/1!, 0;
-40097784/5!, 112770/4!, 4067/3!, -1715/2!, 2401/1!, 0; ...

Cf. A111830, A111834 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111838 (q=8).

PARI
```
T(n,k,q=7)=local(A=Mat(1),B);if(n
				
```

T(n, k) = 7^k*T(n-k, 0) = A111834(n-k) for n>=k>=0.

A111835 Triangle P, read by rows, that satisfies [P^8](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(8*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(0,k)=1 and P(k,k)=1 for all k>=0.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 232, 64, 1, 1, 36968, 16192, 512, 1, 1, 35593832, 21928768, 1047040, 4096, 1, 1, 219379963496, 178379459392, 11424946688, 67096576, 32768, 1, 1, 9003699178010216, 9288403489672000, 748093366229504, 5862250172416

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Also P(n,k) = partitions of (8^n - 8^(n-k)) into powers of 8 <= 8^(n-k).

			Let q=8; 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/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...
and L(x) = x - 6/2!*x^2 + 142/3!*x^3 + 31800/4!*x^4 +... (A111839).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(8*x) + m^3/3!*L(x)*L(8*x)*L(8^2*x) + m^4/4!*L(x)*L(8*x)*L(8^2*x)*L(8^3*x) + ...
Triangle P begins:
1;
1,1;
1,8,1;
1,232,64,1;
1,36968,16192,512,1;
1,35593832,21928768,1047040,4096,1;
1,219379963496,178379459392,11424946688,67096576,32768,1; ...
where P^8 shifts columns left and up one place:
1;
8,1;
232,64,1;
36968,16192,512,1; ...

Cf. A111836 (column 1), A111837 (row sums), A111838 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111825 (q=6), A111830 (q=7).

PARI
```
P(n,k,q=8)=local(A=Mat(1),B);if(n
				
```

Let q=8; 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/(1-x) = Sum_{n>=1} Product_{j=0..n-1} L(q^j*x)/(j+1) and L(x) equals the g.f. of column 0 of the matrix log of P (A111839).

cp's OEIS Frontend

A111839 Column 0 of the matrix logarithm (A111838) of triangle A111835, which shifts columns left and up under matrix 8th power; 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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A111815 Matrix log of triangle A078122, 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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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)!.

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A111823 Matrix log of triangle A111820, which shifts columns left and up under matrix 5th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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A111828 Matrix log of triangle A111825, which shifts columns left and up under matrix 6th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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A111833 Matrix log of triangle A111830, which shifts columns left and up under matrix 7th power; 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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A111835 Triangle P, read by rows, that satisfies [P^8](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(8*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(0,k)=1 and P(k,k)=1 for all k>=0.

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