A107725
Column 0 of A107724, which is the matrix logarithm of triangle A107719.
Original entry on oeis.org
0, 1, 4, 31, 343, 4855, 83209, 1670743, 38436673, 996825703, 28778874481, 915636860023, 31837734460129, 1201360385000071, 48899006799783889, 2135779996980897175, 99646497213608842561, 4946194601237466540967
Offset: 0
-
{a(n)=local(L,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,(-1)^(i-1)*(M-M^0)^i/i);return(if(n<0,0,L[n+1,1]/3))}
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.
Original entry on oeis.org
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
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.
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{T(n,k)=if(n
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{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(""))
Showing 1-2 of 2 results.
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