A111942 -id:A111942 - cp's OEIS frontend

A111941 Matrix log of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -1, -1, 0, 1, 1, 1, 0, -2, -1, -1, -1, 0, 4, 2, 1, 1, 1, 0, -12, -4, -2, -1, -1, -1, 0, 36, 12, 4, 2, 1, 1, 1, 0, -144, -36, -12, -4, -2, -1, -1, -1, 0, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -86400, -14400, -2880, -576, -144, -36

Offset: 0

Paul D. Hanna, Aug 23 2005

			Triangle begins:
0;
1, 0;
-1, -1, 0;
1, 1, 1, 0;
-2, -1, -1, -1, 0;
4, 2, 1, 1, 1, 0;
-12, -4, -2, -1, -1, -1, 0;
36, 12, 4, 2, 1, 1, 1, 0;
-144, -36, -12, -4, -2, -1, -1, -1, 0;
576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
518400, 86400, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-3628800, -518400, -86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0; ...
where, apart from signs, the columns are all the same (A111942).
...
Triangle A111940 begins:
1;
1, 1;
-1, -1, 1;
0, 0, 1, 1;
0, 0, -1, -1, 1;
0, 0, 0, 0, 1, 1;
0, 0, 0, 0, -1, -1, 1;
0, 0, 0, 0, 0, 0, 1 ,1;
0, 0, 0, 0, 0, 0, -1, -1, 1; ...
where the matrix inverse shifts columns left and up one place.
...
The matrix log of A111940, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, -1/1!, 0;
1/3!, 1/2!, 1/1!, 0;
-2/4!, -1/3!, -1/2!, -1/1!, 0;
4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-2880/10!, -576/9!, -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
14400/11!, 2880/10!, 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; ...
Note that the square of the matrix log of A111940 begins:
0;
0, 0;
-1, 0, 0;
0, -1, 0, 0;
-1/12, 0, -1, 0, 0;
0, -1/12, 0, -1, 0, 0;
-1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/16632, 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; ...
where nonzero terms are negative unit fractions with denominators given by A002544:
[1, 12, 90, 560, 3150, 16632, 84084, 411840, ...,  C(2*n+1,n)*(n+1)^2, ...].

Cf. A111940 (triangle), A111942 (column 0), A110504 (variant).

{T(n,k,q=-1) = local(A=Mat(1),B); if(n

T(n, k) = (-1)^k*T(n-k, 0) = (-1)^k*A111942(n-k) for n>=k>=0.

A264638 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 4.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 28, 16, 1, 12, 400, 960, 128, 2, 36, 7776, 107568, 46656, 1152, 4, 144, 229392, 21565440, 71663616, 4561920, 12960, 8, 576, 9082368, 10027929600, 229132800000, 98288640000, 829440000, 269568, 16, 2880, 481406976

Offset: 1

R. H. Hardin, Nov 19 2015

Table starts
..1......1.........2...........4...........12..........36...........144
..1......3........28.........400.........7776......229392.......9082368
..1.....16.......960......107568.....21565440.10027929600.7163873280000
..1....128.....46656....71663616.229132800000
..2...1152...4561920.98288640000
..4..12960.829440000
..8.269568
.16

			Some solutions for n=3 k=4
..4..8..1..9....0..4..8..1....0..4..5..1....0..2..5..3....4..8..6..0
..0..6..2..5...10..2..7..9....2..8.11..9....8.10..1..7...10..9..7..5
.10..7.11..3....6.11..3..5....6.10..3..7....4..6..9.11....2..1.11..3

R. H. Hardin, Table of n, a(n) for n = 1..39

Row 1 is A111942(n+1).

A264560 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 3.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 24, 15, 2, 12, 316, 888, 156, 4, 36, 6080, 84672, 58368, 2032, 8, 144, 149824, 17444160, 66194496, 7019136, 30912, 24, 576, 5728896, 5356808640

Offset: 1

R. H. Hardin, Nov 17 2015

Table starts
..1.....1.......2........4.......12.........36.....144.576
..1.....3......24......316.....6080.....149824.5728896
..1....15.....888....84672.17444160.5356808640
..2...156...58368.66194496
..4..2032.7019136
..8.30912
.24

			Some solutions for n=3 k=4
..4..6..1..9....6..4..9..3....0.11..9..1....0..9..1..3....6..4.10..0
.11..3..7..5....8.10..0..7....6..5..3..7....6..8..4..7....9..1..7..3
..2..0.10..8....2..1..5.11....4..8..2.10...10..2.11..5....8..2.11..5

Row 1 is A111942(n+1).

A264659 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 5.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 16, 8, 1, 12, 148, 314, 38, 1, 36, 2400, 20940, 8944, 223, 2, 144, 62800, 3084352, 5911808, 437568, 2412, 4, 576, 2150784, 726042624, 7707842560, 2436623360, 31791360, 26400, 8, 2880, 89051520, 284038382592

Offset: 1

R. H. Hardin, Nov 20 2015

Table starts
.1.....1........2..........4.........12........36..........144......576.2880
.1.....3.......16........148.......2400.....62800......2150784.89051520
.1.....8......314......20940....3084352.726042624.284038382592
.1....38.....8944....5911808.7707842560
.1...223...437568.2436623360
.2..2412.31791360
.4.26400
.8

			Some solutions for n=3 k=4
..2..0..1..5....8..6.10..5....0.10..7..1....8.10..0.11....2..0..6..5
..8.10..7.11....4..2..0..1...11..5..9..3....9..1..5..7....8.10.11..7
..4..6..3..9....9..7.11..3....2..6..4..8....4..6..2..3....4..3..1..9

Row 1 is A111942(n+1).

A264704 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 6.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 15, 6, 1, 12, 212, 296, 36, 1, 36, 3712, 26304, 9024, 288, 1, 144, 96256, 4689792, 7708608, 492480, 2240, 2, 576, 3423168, 1263257856, 15216574464, 4137827328, 30233088, 20736, 4, 2880, 158521536, 603321053184

Offset: 1

R. H. Hardin, Nov 21 2015

Table starts
.1.....1........2..........4..........12.........36..........144.......576.2880
.1.....3.......15........212........3712......96256......3423168.158521536
.1.....6......296......26304.....4689792.1263257856.603321053184
.1....36.....9024....7708608.15216574464
.1...288...492480.4137827328
.1..2240.30233088
.2.20736

			Some solutions for n=3 k=4
..0..6..8..7....4..6..0..7....8..6..2..0....4..2..6..7....0..6..3..1
..2..4.10..1...10..2..8..1...10..1..5..3...10..8..0..1....8..2.10..7
..9..5.11..3....5.11..3..9....4..7.11..9...11..3..9..5....4..5.11..9

Row 1 is A111942(n+1).

A264794 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 7.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 15, 6, 1, 12, 130, 170, 23, 1, 36, 1944, 9656, 3089, 112, 1, 144, 40188, 1369408, 1685728, 106568, 561, 1, 576, 1149120, 275062400, 1816673024, 410210112, 4676432, 3519, 2, 2880, 49148112, 89203565824

Offset: 1

R. H. Hardin, Nov 25 2015

Table starts
.1....1.......2.........4.........12........36.........144......576.2880
.1....3......15.......130.......1944.....40188.....1149120.49148112
.1....6.....170......9656....1369408.275062400.89203565824
.1...23....3089...1685728.1816673024
.1..112..106568.410210112
.1..561.4676432
.1.3519
.2

			Some solutions for n=3 k=4
..0..1..7..9....0..9..1..7....0..1..9..7....8..7..1..9....9..7..1..3
..4..8..2.10....8..2..3.11....2..4.10..8....2..0..4.10...10..0..2..4
..6.11..3..5....4..6.10..5....6..5..3.11....6..3..5.11....6..8..5.11

Row 1 is A111942(n+1).

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).

A111940 Triangle P, read by rows, that satisfies [P^-1](n,k) = P(n+1,k+1) for n >= k >= 0, with P(k,k)=1 and P(k+1,1)=P(k+1,0) for k >= 0, where [P^-1] denotes the matrix inverse of P.

Original entry on oeis.org

1, 1, 1, -1, -1, 1, 0, 0, 1, 1, 0, 0, -1, -1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1

Offset: 0

Paul D. Hanna, Aug 23 2005

			Triangle P begins:
   1;
   1,  1;
  -1, -1,  1;
   0,  0,  1,  1;
   0,  0, -1, -1,  1;
   0,  0,  0,  0,  1,  1;
   0,  0,  0,  0, -1, -1,  1;
   0,  0,  0,  0,  0,  0,  1,  1;
   0,  0,  0,  0,  0,  0, -1, -1,  1; ...
where P^-1 shifts columns left and up one place:
   1;
  -1,  1;
   0,  1,  1;
   0, -1, -1,  1;
   0,  0,  0,  1,  1;
   0,  0,  0, -1, -1,  1; ...

Cf. A111941 (matrix log), A111942, A110503 (variant).

{P(n,k,q=-1) = local(A=Mat(1),B); if(n

The g.f. of column k of matrix power P^m (ignoring leading zeros) is:

cos(m*arccos(1-x^2/2)) + (-1)^k * sin(m*arccos(1-x^2/2)) * (1-x/2) / sqrt(1-x^2/4).

cp's OEIS Frontend

A111941 Matrix log of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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A264638 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 4.

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A264560 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 3.

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A264659 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 5.

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A264704 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 6.

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A264794 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 7.

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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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A111940 Triangle P, read by rows, that satisfies [P^-1](n,k) = P(n+1,k+1) for n >= k >= 0, with P(k,k)=1 and P(k+1,1)=P(k+1,0) for k >= 0, where [P^-1] denotes the matrix inverse of P.

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