A111547
Column 3 of triangle A111544; also found in column 0 of triangle A111549, which equals the matrix logarithm of A111544.
Original entry on oeis.org
1, 4, 23, 165, 1383, 13083, 136863, 1562715, 19301319, 256191363, 3636036783, 54956529675, 881578601559, 14964805041363, 268105552191423, 5057384615702235, 100224731537223399, 2082402995330965923
Offset: 0
-
{a(n)=if(n<0,0,(matrix(n+4,n+4,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+2)!/2!*x^i)),m-j-1))))^-1)[n+4,4])}
A111550
Column 1 of A111549, which is the matrix log of A111544.
Original entry on oeis.org
0, 2, 6, 32, 222, 1824, 17016, 176112, 1993392, 24438960, 322294896, 4548010032, 68385367152, 1091838106800, 18454096189296, 329306074785072, 6189015238217712, 122232512688657840, 2531600753529542256
Offset: 0
-
{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+2)!/2!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+2,2])}
A111551
Column 2 of A111549, which is the matrix log of A111544.
Original entry on oeis.org
0, 3, 9, 47, 321, 2607, 24099, 247527, 2783331, 33924303, 445016619, 6249234807, 93541817331, 1487200667103, 25037315924859, 445123900236807, 8336458657796931, 164100631571947503, 3388128151043405259
Offset: 0
-
{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+2)!/2!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+3,3])}
A111544
Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+3 of T), or [T^p](m,0) = p*T(p+m,p+3) for all m>=1 and p>=-3.
Original entry on oeis.org
1, 1, 1, 5, 2, 1, 33, 9, 3, 1, 261, 57, 15, 4, 1, 2361, 441, 99, 23, 5, 1, 23805, 3933, 783, 165, 33, 6, 1, 263313, 39249, 7083, 1383, 261, 45, 7, 1, 3161781, 430677, 71415, 13083, 2361, 393, 59, 8, 1, 40907241, 5137641, 789939, 136863, 23805, 3861, 567, 75, 9, 1
Offset: 0
SHIFT_LEFT(column 0 of T^-3) = -3*(column 0 of T);
SHIFT_LEFT(column 0 of T^-2) = -2*(column 1 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 2 of T);
SHIFT_LEFT(column 0 of log(T)) = column 3 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 4 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1,1;
5,2,1;
33,9,3,1;
261,57,15,4,1;
2361,441,99,23,5,1;
23805,3933,783,165,33,6,1;
263313,39249,7083,1383,261,45,7,1;
3161781,430677,71415,13083,2361,393,59,8,1; ...
After initial term, column 2 is 3 times column 0.
Matrix inverse T^-1 = A111548 starts:
1;
-1,1;
-3,-2,1;
-15,-3,-3,1;
-99,-15,-3,-4,1;
-783,-99,-15,-3,-5,1;
-7083,-783,-99,-15,-3,-6,1; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 2 of T.
Matrix logarithm log(T) = A111549 is:
0;
1,0;
4,2,0;
23,6,3,0;
165,32,9,4,0;
1383,222,47,13,5,0;
13083,1824,321,70,18,6,0; ...
compare column 0 of log(T) to column 3 of T.
-
T[n_, k_] := T[n, k] = If[nJean-François Alcover, Aug 09 2018, from PARI *)
-
T(n,k)=if(n
A111560
Matrix logarithm of triangle A111553.
Original entry on oeis.org
0, 1, 0, 5, 2, 0, 34, 7, 3, 0, 282, 44, 10, 4, 0, 2696, 354, 60, 14, 5, 0, 28792, 3328, 470, 84, 19, 6, 0, 337072, 35144, 4344, 654, 118, 25, 7, 0, 4273632, 407984, 45320, 6008, 936, 164, 32, 8, 0, 58195072, 5137824, 521200, 62344, 8704, 1352, 224, 40, 9, 0
Offset: 0
Triangle begins:
0;
1,0;
5,2,0;
34,7,3,0;
282,44,10,4,0;
2696,354,60,14,5,0;
28792,3328,470,84,19,6,0;
337072,35144,4344,654,118,25,7,0;
4273632,407984,45320,6008,936,164,32,8,0; ...
-
{T(n,k)=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+3)!/3!*x^i)),m-j-1))))); sum(i=1,#M,(M^0-M)^i/i)[n+1,k+1]}
Showing 1-5 of 5 results.
Comments