A105627 -id:A105627 - cp's OEIS frontend

A105615 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((2p-1)/2)) = (2p-1)(column p of T), or [T^((2p-1)/2)](m,0) = (2p-1)T(p+m,p+1) for all m>=1 and p>=0.

1, 2, 1, 10, 4, 1, 74, 26, 6, 1, 706, 226, 50, 8, 1, 8162, 2426, 522, 82, 10, 1, 110410, 30826, 6498, 1010, 122, 12, 1, 1708394, 451586, 93666, 14458, 1738, 170, 14, 1, 29752066, 7489426, 1532970, 235466, 28226, 2754, 226, 16, 1, 576037442

Offset: 0

Paul D. Hanna, Apr 16 2005

Column 0 is A000698 (related to double factorials), offset 1. Column 1 is A105616 (column 0 of T^(1/2), offset 1). The matrix logarithm divided by 2 yields the integer triangle A105629.

Compare with triangular matrix A107717, which satisfies: SHIFT_LEFT(column 0 of A107717^((3*k-1)/3)) = (3*k-1)*(column k of A107717).

			SHIFT_LEFT(column 0 of T^(-1/2)) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^(1/2)) = 1*(column 1 of T);
SHIFT_LEFT(column 0 of T^(3/2)) = 3*(column 2 of T);
SHIFT_LEFT(column 0 of T^(5/2)) = 5*(column 3 of T).
Triangle begins:
1;
2,1;
10,4,1;
74,26,6,1;
706,226,50,8,1;
8162,2426,522,82,10,1;
110410,30826,6498,1010,122,12,1;
1708394,451586,93666,14458,1738,170,14,1;
29752066,7489426,1532970,235466,28226,2754,226,16,1; ...
Matrix square-root T^(1/2) is A105623 which begins:
1;
1,1;
4,2,1;
26,10,3,1;
226,74,19,4,1;
2426,706,167,31,5,1; ...
compare column 0 of T^(1/2) to column 1 of T;
also, column 1 of T^(1/2) equals column 0 of T.
Matrix inverse square-root T^(-1/2) is A105620 which begins:
1;
-1,1;
-2,-2,1;
-10,-4,-3,1;
-74,-20,-7,-4,1;
-706,-148,-39,-11,-5,1; ...
compare column 0 of T^(-1/2) to column 0 of T.
Matrix inverse T^-1 is A105619 which begins:
1;
-2,1;
-2,-4,1;
-10,-2,-6,1;
-74,-10,-2,-8,1;
-706,-74,-10,-2,-10,1;
-8162,-706,-74,-10,-2,-12,1; ...

Cf. A000698 (column 0), A105616 (column 1), A105617 (column 2), A105618 (row sums), A105619 (T^-1), A105620 (T^(-1/2)), A105623 (T^(1/2)), A105627 (T^(3/2)), A105629 (matrix log).

Cf. A107717.

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

{T(n,k) = if(n=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", ")); print(""))

T(n, k) = 2*(k+1)*T(n, k+1) + Sum_{j=1..n-k-1} T(j, 0)*T(n, j+k+1) for n>k>=0, with T(n, n) = 1 for n>=0. T(n, 0) = A000698(n+1) for n>=0.

A105626 Triangular matrix T, read by rows, that satisfies T^2 = A105615^3; also equals the matrix cube of triangle A105623.

Original entry on oeis.org

1, 3, 1, 18, 6, 1, 150, 48, 9, 1, 1566, 480, 93, 12, 1, 19494, 5736, 1125, 153, 15, 1, 280998, 79584, 15681, 2190, 228, 18, 1, 4598910, 1256808, 247929, 35181, 3780, 318, 21, 1, 84237246, 22262640, 4389213, 629424, 68961, 6000, 423, 24, 1, 1707637734

Offset: 0

Paul D. Hanna, Apr 16 2005

SHIFT_LEFT(column 0 of T) = 3*(column 2 of A105615). A105623 equals the matrix square-root of triangle A105615.

			Triangle begins:
1;
3,1;
18,6,1;
150,48,9,1;
1566,480,93,12,1;
19494,5736,1125,153,15,1;
280998,79584,15681,2190,228,18,1;
4598910,1256808,247929,35181,3780,318,21,1;
84237246,22262640,4389213,629424,68961,6000,423,24,1; ...

Cf. A105615, A105617, A105623, A105627 (column 1), A105628 (row sums).

T(n,k)=local(R,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-3); R=(M+M^0)/2;for(i=1,floor(2*log(n+2)),R=(R+M*R^(-1))/2); return(if(n

T(n+1, 0) = 3*A105615(n+2, 2) = 3*A105617(n) for n>=0.

cp's OEIS Frontend

A105615 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((2p-1)/2)) = (2p-1)(column p of T), or [T^((2p-1)/2)](m,0) = (2p-1)T(p+m,p+1) for all m>=1 and p>=0.

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A105626 Triangular matrix T, read by rows, that satisfies T^2 = A105615^3; also equals the matrix cube of triangle A105623.

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cp's OEIS Frontend

A105615 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((2*p-1)/2)) = (2*p-1)*(column p of T), or [T^((2*p-1)/2)](m,0) = (2*p-1)*T(p+m,p+1) for all m>=1 and p>=0.

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Author

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Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A105626 Triangular matrix T, read by rows, that satisfies T^2 = A105615^3; also equals the matrix cube of triangle A105623.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A105615 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((2p-1)/2)) = (2p-1)(column p of T), or [T^((2p-1)/2)](m,0) = (2p-1)T(p+m,p+1) for all m>=1 and p>=0.