A111557
Column 4 of triangle A111553; also found in column 0 of triangle A111560, which equals the matrix logarithm of A111553.
Original entry on oeis.org
1, 5, 34, 282, 2696, 28792, 337072, 4273632, 58195072, 846038912, 13072140032, 213897731712, 3695682017792, 67254929193472, 1286282280266752, 25802708552696832, 541894309127053312, 11894387852938452992
Offset: 0
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{a(n)=if(n<0,0,(matrix(n+5,n+5,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))))^-1)[n+5,5])}
A111561
Column 1 of A111560, which is the matrix log of A111553.
Original entry on oeis.org
0, 2, 7, 44, 354, 3328, 35144, 407984, 5137824, 69566144, 1006372864, 15481560064, 252326173824, 4344015683584, 78793242566144, 1502374856981504, 30052439647908864, 629485600310349824, 13783057303819485184
Offset: 0
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{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+3)!/3!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+2,2])}
A111562
Column 2 of A111560, which is the matrix log of A111553.
Original entry on oeis.org
0, 3, 10, 60, 470, 4344, 45320, 521200, 6513280, 87613440, 1260188800, 19286781440, 312884243840, 5363632995840, 96904980170240, 1840977689943040, 36700487712701440, 766296476640215040, 16728728381231472640
Offset: 0
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{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+3)!/3!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+3,3])}
A111553
Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+4 of T), or [T^p](m,0) = p*T(p+m,p+4) for all m>=1 and p>=-4.
Original entry on oeis.org
1, 1, 1, 6, 2, 1, 46, 10, 3, 1, 416, 72, 16, 4, 1, 4256, 632, 116, 24, 5, 1, 48096, 6352, 1016, 184, 34, 6, 1, 591536, 70912, 10176, 1664, 282, 46, 7, 1, 7840576, 864192, 113216, 17024, 2696, 416, 60, 8, 1, 111226816, 11371072, 1375456, 192384, 28792, 4256, 592, 76, 9, 1
Offset: 0
SHIFT_LEFT(column 0 of T^-4) = -4*(column 0 of T);
SHIFT_LEFT(column 0 of T^-3) = -3*(column 1 of T);
SHIFT_LEFT(column 0 of T^-2) = -2*(column 2 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 3 of T);
SHIFT_LEFT(column 0 of log(T)) = column 4 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 5 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1,1;
6,2,1;
46,10,3,1;
416,72,16,4,1;
4256,632,116,24,5,1;
48096,6352,1016,184,34,6,1;
591536,70912,10176,1664,282,46,7,1;
7840576,864192,113216,17024,2696,416,60,8,1; ...
After initial term, column 3 is 4 times column 0.
Matrix inverse T^-1 = A111559 starts:
1;
-1,1;
-4,-2,1;
-24,-4,-3,1;
-184,-24,-4,-4,1;
-1664,-184,-24,-4,-5,1;
-17024,-1664,-184,-24,-4,-6,1; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 3 of T.
Matrix logarithm log(T) = A111560 is:
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; ...
compare column 0 of log(T) to column 4 of T.
Cf.
A111531 (column 0),
A111554 (column 1),
A111555 (column 2),
A111556 (column 3),
A111557 (column 4),
A111558 (row sums),
A111559 (matrix inverse),
A111560 (matrix log); related tables:
A111528,
A104980,
A111536,
A111544.
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T[n_, k_] := T[n, k] = If[nJean-François Alcover, Aug 09 2018, from PARI *)
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T(n,k)=if(n
Showing 1-4 of 4 results.
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