A111835 -id:A111835 - cp's OEIS frontend

A111838 Matrix log of triangle A111835, which shifts columns left and up under matrix 8th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -6, 8, 0, 142, -48, 64, 0, 31800, 1136, -384, 512, 0, -159468264, 254400, 9088, -3072, 4096, 0, -2481298801008, -1275746112, 2035200, 72704, -24576, 32768, 0, 1414130111428687344, -19850390408064, -10205968896, 16281600, 581632, -196608, 262144, 0

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 8^k multiplied by column 0 (A111839) when ignoring zeros above the diagonal.

			Matrix log of A111835, with factorial denominators, begins:
0;
1/1!, 0;
-6/2!, 8/1!, 0;
142/3!, -48/2!, 64/1!, 0;
31800/4!, 1136/3!, -384/2!, 512/1!, 0;
-159468264/5!, 254400/4!, 9088/3!, -3072/2!, 4096/1!, 0; ...

Cf. A111835, A111839 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7).

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

T(n, k) = 8^k*T(n-k, 0) = A111839(n-k) for n>=k>=0.

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!.

Original entry on oeis.org

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).

A111836 Number of partitions of 7*8^n into powers of 8, also equals column 1 of triangle A111835, which shifts columns left and up under matrix 8th power.

Original entry on oeis.org

1, 8, 232, 36968, 35593832, 219379963496, 9003699178010216, 2530260913162860295784, 4970141819535151534947497576, 69322146154435681317709098939119208

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Let q=8; a(n) equals the partitions of (q-1)*q^n into powers of q, or, the coefficient of x^((q-1)*q^n) in 1/Product_{j>=0}(1-x^(q^j)).

Alois P. Heinz, Table of n, a(n) for n = 0..40

Cf. A111835 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111821 (q=5), A111826 (q=6), A111831 (q=7).

a(n,q=8)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(A[n+2,2]))

a(n) = [x^(7*8^n)] 1/Product_{j>=0}(1-x^(8^j)).

A111837 Number of partitions of 8^n into powers of 8, also equals the row sums of triangle A111835, which shifts columns left and up under matrix 8th power.

Original entry on oeis.org

1, 2, 10, 298, 53674, 58573738, 409251498922, 19046062579215274, 6071277235712979102634, 13531779463193107731083553706, 214224474679766323250278564215516074, 24390479071277895100812271376578637910371242, 20173309182842708837666031701435147789403500172143530

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..45

Cf. A111835, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111822 (q=5), A111827 (q=6), A111832 (q=7). Column 8 of A145515.

a(n,q=8)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(sum(k=0,n,A[n+1,k+1])))

a(n) = [x^(8^n)] 1/Product_{j>=0} (1-x^(8^j)).

A111825 Triangle P, read by rows, that satisfies [P^6](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(6*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, 6, 1, 1, 96, 36, 1, 1, 6306, 3816, 216, 1, 1, 1883076, 1625436, 139536, 1296, 1, 1, 2700393702, 3121837776, 360839016, 5036256, 7776, 1, 1, 19324893252552, 28794284803908, 4200503990976, 78293629296, 181382976, 46656, 1

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Also P(n,k) = the partitions of (6^n - 6^(n-k)) into powers of 6 <= 6^(n-k).

			Let q=6; 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 - 4/2!*x^2 + 42/3!*x^3 + 7296/4!*x^4 +... (A111829).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(6*x) + m^3/3!*L(x)*L(6*x)*L(6^2*x) +
m^4/4!*L(x)*L(6*x)*L(6^2*x)*L(6^3*x) + ...
Triangle P begins:
1;
1,1;
1,6,1;
1,96,36,1;
1,6306,3816,216,1;
1,1883076,1625436,139536,1296,1;
1,2700393702,3121837776,360839016,5036256,7776,1; ...
where P^6 shifts columns left and up one place:
1;
6,1;
96,36,1;
6306,3816,216,1; ...

Cf. A111826 (column 1), A111827 (row sums), A111828 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111830 (q=7), A111835 (q=8).

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

Let q=6; 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 (A111829).

A111820 Triangle P, read by rows, that satisfies [P^5](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(5*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, 5, 1, 1, 55, 25, 1, 1, 2055, 1525, 125, 1, 1, 291430, 311525, 38875, 625, 1, 1, 165397680, 239305275, 40338875, 975625, 3125, 1, 1, 390075741430, 735920617775, 157056792000, 5077475625, 24409375, 15625, 1

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Also P(n,k) = the partitions of (5^n - 5^(n-k)) into powers of 5 <= 5^(n-k).

			Let q=5; 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 - 3/2!*x^2 + 16/3!*x^3 + 2814/4!*x^4 +... (A111824).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(5*x) + m^3/3!*L(x)*L(5*x)*L(5^2*x) +
m^4/4!*L(x)*L(5*x)*L(5^2*x)*L(5^3*x) + ...
Triangle P begins:
1;
1,1;
1,5,1;
1,55,25,1;
1,2055,1525,125,1;
1,291430,311525,38875,625,1;
1,165397680,239305275,40338875,975625,3125,1; ...
where P^5 shifts columns left and up one place:
1;
5,1;
55,25,1;
2055,1525,125,1;
291430,311525,38875,625,1; ...

Cf. A111821 (column 1), A111822 (row sums), A111823 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111825 (q=6), A111830 (q=7), A111835 (q=8).

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

Let q=5; 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 (A111824).

A111830 Triangle P, read by rows, that satisfies [P^7](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(7*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, 7, 1, 1, 154, 49, 1, 1, 16275, 8281, 343, 1, 1, 9106461, 6558209, 410914, 2401, 1, 1, 28543862991, 27307109501, 2298650515, 20170801, 16807, 1, 1, 521136519414483, 636922972420469, 67522139062441, 790856748801, 988621354

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Also P(n,k) = partitions of (7^n - 7^(n-k)) into powers of 7 <= 7^(n-k).

			Let q=7; 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 - 5/2!*x^2 + 83/3!*x^3 + 16110/4!*x^4 +... (A111834).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(7*x) + m^3/3!*L(x)*L(7*x)*L(7^2*x) +
m^4/4!*L(x)*L(7*x)*L(7^2*x)*L(7^3*x) + ...
Triangle P begins:
1;
1,1;
1,7,1;
1,154,49,1;
1,16275,8281,343,1;
1,9106461,6558209,410914,2401,1;
1,28543862991,27307109501,2298650515,20170801,16807,1; ...
where P^7 shifts columns left and up one place:
1;
7,1;
154,49,1;
16275,8281,343,1; ...

Cf. A111831 (column 1), A111832 (row sums), A111833 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111825 (q=6), A111835 (q=8).

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

Let q=7; 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 (A111834).

cp's OEIS Frontend

A111838 Matrix log of triangle A111835, which shifts columns left and up under matrix 8th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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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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A111836 Number of partitions of 7*8^n into powers of 8, also equals column 1 of triangle A111835, which shifts columns left and up under matrix 8th power.

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A111837 Number of partitions of 8^n into powers of 8, also equals the row sums of triangle A111835, which shifts columns left and up under matrix 8th power.

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A111825 Triangle P, read by rows, that satisfies [P^6](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(6*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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A111820 Triangle P, read by rows, that satisfies [P^5](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(5*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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A111830 Triangle P, read by rows, that satisfies [P^7](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(7*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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