A111834 -id:A111834 - cp's OEIS frontend

A111814 Column 0 of the matrix logarithm (A111813) of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

0, 1, 0, -2, 0, 216, 0, -568464, 0, 36058658688, 0, -53694310935340800, 0, 1790669979087018171448320, 0, -1280832788659041410080025283840000, 0, 18961468161294510864200732026858464699187200, 0

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Surprisingly, the e.g.f. A(x) is an odd function: A(-x) = -A(x). Let q=2; the g.f. of column k of A078121^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 - 2/3!*x^3 + 216/5!*x^5 - 568464/7!*x^7 + ...
where A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(2*x)/2! + A(x)*A(2*x)*A(2^2*x)/3!
+ A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! + ...
also:
x/(1-x^2) = A(x) + A(x)*A(2*x)*A(2^2*x)/3!
+ A(x)*A(2*x)*A(2^2*x)*A(2^3*x)*A(2^4*x)/5! + ...
Let G(x) be the g.f. of A002577 (column 1 of A078121), then
G(x) = 1 + 2*A(x) + 2^2*A(x)*A(2*x)/2! +
2^3*A(x)*A(2*x)*A(2^2*x)/3! +
2^4*A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! + ...

Paul D. Hanna, Table of n, a(n), n== 0..50.

Cf. A078121 (triangle), A002577, A111813 (matrix log); A110505 (q=-1), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

{a(n,q=2)=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(2^j*x)/(j+1). E.g.f. satisfies: x/(1-x^2) = Sum_{n>=1}Prod_{j=0..2*n}A(2^j*x)/(j+1).

A111816 Column 0 of the matrix logarithm (A111815) of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, -1, -3, 150, 1236, -2555748, -64342116, 5885700899760, 442646611978752, -1737387344860364226240, -367706581563500487774720, 60788555325888838346137808787840, 34626906551623392401873575206240000, -237458311254822429335982538087618909465992960

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Let q=3; the g.f. of column k of A078122^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 - 1/2!*x^2 - 3/3!*x^3 + 150/4!*x^4 + 1236/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3! +
A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
Let G(x) be the g.f. of A078124 (column 1 of A078122), then
G(x) = 1 + 3*A(x) + 3^2*A(x)*A(3*x)/2! +
3^3*A(x)*A(3*x)*A(3^2*x)/3! +
3^4*A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...

Cf. A078122 (triangle), A078124, A111815 (matrix log); A110505 (q=-1), A111814 (q=2), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

{a(n,q=3)=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(3^j*x)/(j+1).

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!.

Original entry on oeis.org

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).

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).

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.

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).

A111819 Column 0 of the matrix logarithm (A111818) of triangle A078536, 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, -2, 2, 840, -76056, -158761104, 390564896784, 14713376473366656, -783793232940393380736, -571732910947761663424746240, 603368029500890443054004423520000, 8390120127886533420753746115877557580800

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Let q=4; the g.f. of column k of A078536^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 - 2/2!*x^2 + 2/3!*x^3 + 840/4!*x^4 - 76056/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-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! + ...
Let G(x) be the g.f. of A111817 (column 1 of A078536), then
G(x) = 1 + 4*A(x) + 4^2*A(x)*A(4*x)/2! +
4^3*A(x)*A(4*x)*A(4^2*x)/3! +
4^4*A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...

Cf. A078536 (triangle), A111817, A111818 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

{a(n,q=4)=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(4^j*x)/(j+1).

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

A111814 Column 0 of the matrix logarithm (A111813) of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

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A111816 Column 0 of the matrix logarithm (A111815) of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.

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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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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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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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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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A111819 Column 0 of the matrix logarithm (A111818) of triangle A078536, 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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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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