A111538 -id:A111538 - cp's OEIS frontend

A111536 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p(column p+2 of T), or [T^p](m,0) = pT(p+m,p+2) for all m>=1 and p>=-2.

1, 1, 1, 4, 2, 1, 22, 8, 3, 1, 148, 44, 14, 4, 1, 1156, 296, 84, 22, 5, 1, 10192, 2312, 600, 148, 32, 6, 1, 99688, 20384, 4908, 1156, 242, 44, 7, 1, 1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1, 12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1

Offset: 0

Paul D. Hanna, Aug 06 2005

Column 0 equals A111529 (related to log of factorial series).

Column 2 (A111538) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111541.

			SHIFT_LEFT(column 0 of T^-2) = -2*(column 0 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 1 of T);
SHIFT_LEFT(column 0 of log(T)) = column 2 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 3 of T);
SHIFT_LEFT(column 0 of T^2) = 2*(column 4 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1, 1;
4, 2, 1;
22, 8, 3, 1;
148, 44, 14, 4, 1;
1156, 296, 84, 22, 5, 1;
10192, 2312, 600, 148, 32, 6, 1;
99688, 20384, 4908, 1156, 242, 44, 7, 1;
1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1;
12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1; ...
...
After initial term, column 1 is twice column 0.
Matrix inverse T^-1 = A111540 starts:
1;
-1, 1;
-2, -2, 1;
-8, -2, -3, 1;
-44, -8, -2, -4, 1;
-296, -44, -8, -2, -5, 1;
-2312, -296, -44, -8, -2, -6, 1;
-20384, -2312, -296, -44, -8, -2, -7, 1;
-199376, -20384, -2312, -296, -44, -8, -2, -8, 1; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 1 of T.
Matrix logarithm log(T) = A111541 is:
0;
1, 0;
3, 2, 0;
14, 5, 3, 0;
84, 22, 8, 4, 0;
600, 128, 36, 12, 5, 0;
4908, 896, 212, 58, 17, 6, 0;
44952, 7220, 1496, 360, 90, 23, 7, 0;
454344, 65336, 12128, 2652, 602, 134, 30, 8, 0;
5016768, 653720, 110288, 22320, 4736, 974, 192, 38, 9, 0; ...
compare column 0 of log(T) to column 2 of T.

Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.

Cf. A111537 (column 1), A111538 (column 2), A111539 (row sums), A111540 (matrix inverse), A111541 (matrix log); related tables: A111528, A104980, A111544, A111553.

T[n_, k_] := T[n, k] = If[nJean-François Alcover, Jan 24 2017, adapted from PARI *)

PARI
```
T(n,k)=if(n
				
```

T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+1, 1)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+2, 1) = 2*T(n+1, 0), T(n+3, 3) = T(n+1, 0), for n>=0.

A111541 Matrix logarithm of triangle A111536.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 14, 5, 3, 0, 84, 22, 8, 4, 0, 600, 128, 36, 12, 5, 0, 4908, 896, 212, 58, 17, 6, 0, 44952, 7220, 1496, 360, 90, 23, 7, 0, 454344, 65336, 12128, 2652, 602, 134, 30, 8, 0, 5016768, 653720, 110288, 22320, 4736, 974, 192, 38, 9, 0, 60062352, 7155104

Offset: 0

Paul D. Hanna, Aug 06 2005

Column 0 (A111538) is found in column 2 of triangle A111536.

			Triangle begins:
0;
1,0;
3,2,0;
14,5,3,0;
84,22,8,4,0;
600,128,36,12,5,0;
4908,896,212,58,17,6,0;
44952,7220,1496,360,90,23,7,0;
454344,65336,12128,2652,602,134,30,8,0;
5016768,653720,110288,22320,4736,974,192,38,9,0; ...

Cf. A111536, A111538 (column 0), A111542 (column 1), A111543 (column 2).

T(n,k)=local(M=matrix(n+1,n+1,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+1)!/1!*x^i)),m-j-1))))); return(if(n

A111542 Column 1 of triangle A111541, which is the matrix logarithm of A111536.

Original entry on oeis.org

0, 2, 5, 22, 128, 896, 7220, 65336, 653720, 7155104, 84998768, 1089232160, 14981704736, 220233312896, 3447195190592, 57261708795776, 1006401042534272, 18663532970127872, 364283224523605760, 7466218532765196800

Offset: 0

Paul D. Hanna, Aug 06 2005

nonn

Column 0 of triangle A111541 is A111536.

Cf. A111541 (triangle), A111538 (column 0), A111543 (column 2), A111536.

{a(n)=local(M=matrix(n+2,n+2,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+1)!/1!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+2,2])}

A111543 Column 2 of triangle A111541, which is the matrix logarithm of A111536.

Original entry on oeis.org

0, 3, 8, 36, 212, 1496, 12128, 110288, 1108064, 12171872, 145061120, 1864321472, 25710635648, 378871778432, 5943632568320, 98936446059776, 1742232571097600, 32367994818881024, 632845309575139328, 12991224275641441280

Offset: 0

Paul D. Hanna, Aug 06 2005

nonn

Cf. A111541 (triangle), A111538 (column 0), A111542 (column 1), A111536.

{a(n)=local(M=matrix(n+3,n+3,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+1)!/1!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+3,3])}

cp's OEIS Frontend

A111536 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p(column p+2 of T), or [T^p](m,0) = pT(p+m,p+2) for all m>=1 and p>=-2.

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A111541 Matrix logarithm of triangle A111536.

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A111542 Column 1 of triangle A111541, which is the matrix logarithm of A111536.

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A111543 Column 2 of triangle A111541, which is the matrix logarithm of A111536.

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cp's OEIS Frontend

A111536 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+2 of T), or [T^p](m,0) = p*T(p+m,p+2) for all m>=1 and p>=-2.

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A111541 Matrix logarithm of triangle A111536.

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A111542 Column 1 of triangle A111541, which is the matrix logarithm of A111536.

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A111543 Column 2 of triangle A111541, which is the matrix logarithm of A111536.

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PARI

A111536 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p(column p+2 of T), or [T^p](m,0) = pT(p+m,p+2) for all m>=1 and p>=-2.