A107727 -id:A107727 - cp's OEIS frontend

A107717 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((3p-1)/3)) = (3p-1)(column p of T), or [T^((3p-1)/3)](m,0) = (3p-1)T(p+m,p) for all m>=1 and p>=0.

1, 3, 1, 21, 6, 1, 219, 57, 9, 1, 2973, 723, 111, 12, 1, 49323, 11361, 1713, 183, 15, 1, 964173, 212151, 31575, 3351, 273, 18, 1, 21680571, 4584081, 675489, 71391, 5799, 381, 21, 1, 551173053, 112480887, 16442823, 1732881, 140529, 9219, 507, 24, 1

Offset: 0

Paul D. Hanna, May 30 2005

Column 0 is A107716 (INVERTi of triple factorials). Column 1 is A107718 (twcie column 0 of T^(2/3), offset 1). The matrix logarithm divided by 3 yields the integer triangle A107724.

			SHIFT_LEFT(column 0 of T^(p-1/3)) = (3*p-1)*(column p of T):
SHIFT_LEFT(column 0 of T^(-1/3)) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^(2/3)) = 2*(column 1 of T);
SHIFT_LEFT(column 0 of T^(5/3)) = 5*(column 2 of T).
Triangle begins:
1;
3,1;
21,6,1;
219,57,9,1;
2973,723,111,12,1;
49323,11361,1713,183,15,1;
964173,212151,31575,3351,273,18,1;
21680571,4584081,675489,71391,5799,381,21,1; ...
Matrix power (2/3), T^(2/3), is A107719 and begins:
1;
2,1;
12,4,1;
114,32,6,1;
1446,364,62,8,1;
22722,5276,854,102,10,1; ...
compare column 0 of T^(2/3) to 2*(column 1 of T).
Matrix inverse cube-root T^(-1/3) is A107727 and begins:
1;
-1,1;
-3,-2,1;
-21,-7,-3,1;
-219,-53,-13,-4,1;
-2973,-583,-115,-21,-5,1; ...
compare column 0 of T^(-1/3) to column 0 of T.
Matrix inverse is A107726 and begins:
1;
-3,1;
-3,-6,1;
-21,-3,-9,1;
-219,-21,-3,-12,1;
-2973,-219,-21,-3,-15,1; ...
compare column 0 of T^(-1) to column 0 of T.

Cf. A105615, A107716, A107718-A107727.

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

{T(n,k)=if(n=j,if(m==j,1,if(m==j+1,-3*j,-T(m-j-1,0)))))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", ")); print(""))

T(n, k) = 3*(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) = A107716(n+1) for n>=0.

A107719 Matrix cube-root of triangle A107717.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 43, 13, 3, 1, 509, 131, 25, 4, 1, 7579, 1741, 303, 41, 5, 1, 135341, 28451, 4681, 587, 61, 6, 1, 2813851, 549757, 87039, 10449, 1011, 85, 7, 1, 66733853, 12247211, 1885177, 220023, 20445, 1603, 113, 8, 1, 1778159275, 308953453

Offset: 0

Paul D. Hanna, May 30 2005

Column 0 is A107720. Column 1 is A107721. Matrix inverse is A107727.

			Triangle begins:
1;
1,1;
5,2,1;
43,13,3,1;
509,131,25,4,1;
7579,1741,303,41,5,1;
135341,28451,4681,587,61,6,1;
2813851,549757,87039,10449,1011,85,7,1; ...

Cf. A107717, A107720, A107721, A107727.

T(n,k)=local(E,L,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-3*j, -polcoeff(1-(1+sum(c=1,m-j,prod(i=0,c-1,3*i+1)*x^c)+x*O(x^(m-j)))^-1,m-j);))))^-1); L=matrix(#M,#M,r,c,if(r>=c,sum(i=1,#M,(-1)^(i-1)*(M-M^0)^i/i)[r,c])); E=matrix(#L,#L,r,c,if(r>=c,sum(i=0,#L,L^i/3^i/i!)[r,c])); if(n

A107728 Matrix inverse of A107722.

Original entry on oeis.org

1, -2, 1, -4, -4, 1, -26, -8, -6, 1, -262, -52, -14, -8, 1, -3482, -524, -102, -22, -10, 1, -56902, -6964, -1130, -184, -32, -12, 1, -1099514, -113804, -16326, -2304, -306, -44, -14, 1, -24494422, -2199028, -287882, -37224, -4326, -476, -58, -16, 1, -617906906, -48988844, -5969382, -727928, -78114

Offset: 0

Paul D. Hanna, May 30 2005

Column 0 shift left = -2*A107721, where A107721 = column 1 of A107719. Column 1 shift left = 2*(column 0) shift left. Matrix square of A107727.

			Triangle begins:
1;
-2,1;
-4,-4,1;
-26,-8,-6,1;
-262,-52,-14,-8,1;
-3482,-524,-102,-22,-10,1;
-56902,-6964,-1130,-184,-32,-12,1;
-1099514,-113804,-16326,-2304,-306,-44,-14,1; ...

Cf. A107719, A107721, A107722, A107727.

{T(n,k)=local(L,N,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-3*j, polcoeff(1/sum(i=0,m-j,prod(r=0,i-1,3*r+1)*x^i)+O(x^m),m-j)))))^-1); L=sum(i=1,#M,(M^0-M)^i/i)/3;N=sum(i=0,#L,L^i/i!); return(if(n<0,0,(N^2)[n+1,k+1]))}

cp's OEIS Frontend

A107717 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((3p-1)/3)) = (3p-1)(column p of T), or [T^((3p-1)/3)](m,0) = (3p-1)T(p+m,p) for all m>=1 and p>=0.

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A107719 Matrix cube-root of triangle A107717.

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A107728 Matrix inverse of A107722.

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cp's OEIS Frontend

A107717 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((3*p-1)/3)) = (3*p-1)*(column p of T), or [T^((3*p-1)/3)](m,0) = (3*p-1)*T(p+m,p) for all m>=1 and p>=0.

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PARI

PARI

Formula

A107719 Matrix cube-root of triangle A107717.

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PARI

A107728 Matrix inverse of A107722.

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PARI

A107717 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((3p-1)/3)) = (3p-1)(column p of T), or [T^((3p-1)/3)](m,0) = (3p-1)T(p+m,p) for all m>=1 and p>=0.