A111821 -id:A111821 - cp's OEIS frontend

A111824 Column 0 of the matrix logarithm (A111823) of triangle A111820, which shifts columns left and up under matrix 5th power; these terms are the result of multiplying the element in row n by n!.

0, 1, -3, 16, 2814, -1092180, -3603928080, 58978973128440, 5974833380453777520, -3294186866481455009752320, -10279982482873484428390722523200, 175129088125361734252730927280177244800

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Let q=5; the g.f. of column k of A111820^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 - 3/2!*x^2 + 16/3!*x^3 + 2814/4!*x^4 - 1092180/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(5*x)/2! + A(x)*A(5*x)*A(5^2*x)/3! +
A(x)*A(5*x)*A(5^2*x)*A(5^3*x)/4! + ...
Let G(x) be the g.f. of A111821 (column 1 of A111820), then
G(x) = 1 + 5*A(x) + 5^2*A(x)*A(5*x)/2! +
5^3*A(x)*A(5*x)*A(5^2*x)/3! +
5^4*A(x)*A(5*x)*A(5^2*x)*A(5^3*x)/4! + ...

Cf. A111820 (triangle), A111821, A111823 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111829 (q=6), A111834 (q=7), A111839 (q=8).

{a(n,q=5)=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(5^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).

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)).

A111831 Number of partitions of 6*7^n into powers of 7, also equals column 1 of triangle A111830, which shifts columns left and up under matrix 7th power.

Original entry on oeis.org

1, 7, 154, 16275, 9106461, 28543862991, 521136519414483, 56980036448207052005, 38084892600214854893482284, 158081960770204032330986210466109, 4125860571927530263431055188002578191656

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Let q=7; 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. A111830 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111821 (q=5), A111826 (q=6), A111836 (q=8).

a(n,q=7)=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^(6*7^n)] 1/Product_{j>=0}(1-x^(7^j)).

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)).

A111817 Number of partitions of 3*4^n into powers of 4, also equals column 1 of triangle A078536, which shifts columns left and up under matrix 4th power.

Original entry on oeis.org

1, 4, 28, 524, 29804, 5423660, 3276048300, 6744720496300, 48290009081437356, 1221415413140406958252, 110523986015743458745929900, 36150734459755630877180158951596

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Let q=4; 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. A078536 (triangle), A002577 (q=2), A078124 (q=3), A111821 (q=5), A111826 (q=6), A111831 (q=7), A111836 (q=8).

a(n,q=4)=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^(3*4^n)] 1/Product_{j>=0}(1-x^(4^j)).

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

Paul D. Hanna, Aug 23 2005

sign

Let q=4; the g.f. of column k of A111845^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

			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! +...

Cf. A111848 (matrix log), A111845 (triangle), A111846, A111821 (variant), A111942 (q=-1), A111811 (q=2), A111844 (q=3).

{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]))}

E.g.f. satisfies: x = Sum_{n>=1} -(-1)^n/n!*Prod_{j=0..n-1} A(4^j*x).

cp's OEIS Frontend

A111824 Column 0 of the matrix logarithm (A111823) of triangle A111820, which shifts columns left and up under matrix 5th 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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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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A111831 Number of partitions of 6*7^n into powers of 7, also equals column 1 of triangle A111830, which shifts columns left and up under matrix 7th power.

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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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A111817 Number of partitions of 3*4^n into powers of 4, also equals column 1 of triangle A078536, which shifts columns left and up under matrix 4th power.

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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!.

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