A105629 -id:A105629 - cp's OEIS frontend

A105630 Column 0 of triangle A105629, which is the matrix logarithm of triangle A105623.

0, 1, 3, 17, 135, 1353, 16251, 226857, 3605775, 64288209, 1270969971, 27603549057, 653517822615, 16755529944729, 462601460800491, 13685474246611737, 431948067953729055, 14489465807596684449, 514794897939436455651

Offset: 0

Paul D. Hanna, Apr 16 2005

nonn

A105623 equals the matrix square-root of triangle A105615.

Cf. A105629, A105623, A105615.

{a(n)=local(L,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)))))^-1); L=sum(i=1,#M,(-1)^(i-1)*(M-M^0)^i/i); return(if(n<0,0,L[n+1,1]/2))}

A105631 Row sums of triangle A105629, which is the matrix logarithm of triangle A105623.

Original entry on oeis.org

0, 1, 5, 27, 195, 1833, 21125, 286451, 4453859, 78031153, 1520668645, 32631020011, 764640901539, 19431070911513, 532315915981605, 15640217893829891, 490640673319698179, 16368499360721508833, 578693283962999831365

Offset: 0

Paul D. Hanna, Apr 16 2005

nonn

A105623 equals the matrix square-root of triangle A105615.

Cf. A105629, A105623, A105615.

{a(n)=local(L,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)))))^-1); L=sum(i=1,#M,(-1)^(i-1)*(M-M^0)^i/i); return(if(n<0,0,sum(k=0,n,L[n+1,k+1])/2))}

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A105630 Column 0 of triangle A105629, which is the matrix logarithm of triangle A105623.

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A105631 Row sums of triangle A105629, which is the matrix logarithm of triangle A105623.

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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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cp's OEIS Frontend

A105630 Column 0 of triangle A105629, which is the matrix logarithm of triangle A105623.

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PARI

A105631 Row sums of triangle A105629, which is the matrix logarithm of triangle A105623.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

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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PARI

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.