A111825 -id:A111825 - cp's OEIS frontend

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)!.

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.

A111829 Column 0 of the matrix logarithm (A111828) of triangle A111825, which shifts columns left and up under matrix 6th power; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, -4, 42, 7296, -7931976, -45557382240, 3064554175021200, 801993619807364206080, -2618439032548254776387771520, -30580166025709706974876961026475520, 4440597519115996836838709580481861376121600

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Let q=6; 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 - 4/2!*x^2 + 42/3!*x^3 + 7296/4!*x^4 - 7931976/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(6*x)/2! + A(x)*A(6*x)*A(6^2*x)/3! +
A(x)*A(6*x)*A(6^2*x)*A(6^3*x)/4! + ...
Let G(x) be the g.f. of A111826 (column 1 of A111825), then
G(x) = 1 + 6*A(x) + 6^2*A(x)*A(6*x)/2! +
6^3*A(x)*A(6*x)*A(6^2*x)/3! +
6^4*A(x)*A(6*x)*A(6^2*x)*A(6^3*x)/4! + ...

Cf. A111825 (triangle), A111826, A111828 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111834 (q=7), A111839 (q=8).

{a(n,q=6)=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(6^j*x)/(j+1).

A111826 Number of partitions of 5*6^n into powers of 6, also equals column 1 of triangle A111825, which shifts columns left and up under matrix 6th power.

Original entry on oeis.org

1, 6, 96, 6306, 1883076, 2700393702, 19324893252552, 709398600017820522, 136347641698786289641932, 139389318443495655514432423662, 767442745549858935398537400096197328

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Let q=6; 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. A111825 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111821 (q=5), A111831 (q=7), A111836 (q=8).

a(n,q=6)=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^(5*6^n)] 1/Product_{j>=0}(1-x^(6^j)).

A111827 Number of partitions of 6^n into powers of 6, also equals the row sums of triangle A111825, which shifts columns left and up under matrix 6th power.

Original entry on oeis.org

1, 2, 8, 134, 10340, 3649346, 6188114528, 52398157106366, 2277627698797283420, 518758596372421679994170, 628925760908337480420110203736, 4109478867142143642923124190955500214

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..52

Cf. A111825, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111822 (q=5), A111832 (q=7), A111837 (q=8). Column 6 of A145515.

a(n,q=6)=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^(6^n)] 1/Product_{j>=0}(1-x^(6^j)).

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).

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).

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

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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A111829 Column 0 of the matrix logarithm (A111828) of triangle A111825, which shifts columns left and up under matrix 6th power; these terms are the result of multiplying the element in row n by n!.

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A111826 Number of partitions of 5*6^n into powers of 6, also equals column 1 of triangle A111825, which shifts columns left and up under matrix 6th power.

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A111827 Number of partitions of 6^n into powers of 6, also equals the row sums of triangle A111825, which shifts columns left and up under matrix 6th power.

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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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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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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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