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A111536
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Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+2 of T), or [T^p](m,0) = p*T(p+m,p+2) for all m>=1 and p>=-2.
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9
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1, 1, 1, 4, 2, 1, 22, 8, 3, 1, 148, 44, 14, 4, 1, 1156, 296, 84, 22, 5, 1, 10192, 2312, 600, 148, 32, 6, 1, 99688, 20384, 4908, 1156, 242, 44, 7, 1, 1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1, 12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Column 0 equals A111529 (related to log of factorial series). Column 2 (A111538) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111541.
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FORMULA
| T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+1, 1)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+2, 1) = 2*T(n+1, 0), T(n+3, 3) = T(n+1, 0), for n>=0.
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EXAMPLE
| SHIFT_LEFT(column 0 of T^-2) = -2*(column 0 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 1 of T);
SHIFT_LEFT(column 0 of log(T)) = column 2 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 3 of T);
SHIFT_LEFT(column 0 of T^2) = 2*(column 4 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1,1;
4,2,1;
22,8,3,1;
148,44,14,4,1;
1156,296,84,22,5,1;
10192,2312,600,148,32,6,1;
99688,20384,4908,1156,242,44,7,1;
1069168,199376,44952,10192,2084,372,58,8,1; ...
After initial term, column 1 is twice column 0.
Matrix inverse T^-1 = A111540 starts:
1;
-1,1;
-2,-2,1;
-8,-2,-3,1;
-44,-8,-2,-4,1;
-296,-44,-8,-2,-5,1;
-2312,-296,-44,-8,-2,-6,1; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 1 of T.
Matrix logarithm log(T) = A111541 is:
0;
1,0;
3,2,0;
14,5,3,0;
84,22,8,4,0;
600,128,36,12,5,0;
4908,896,212,58,17,6,0; ...
compare column 0 of log(T) to column 2 of T.
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PROG
| (PARI) {T(n, k)=if(n<k|k<0, 0, if(n==k, 1, if(n==k+1, n, k*T(n, k+1)+sum(j=0, n-k-1, T(j+1, 1)*T(n, j+k+1)))))}
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CROSSREFS
| Cf. A111537 (column 1), A111538 (column 2), A111539 (row sums), A111540 (matrix inverse), A111541 (matrix log); related tables: A111528, A104980, A111544, A111553.
Sequence in context: A152391 A144088 A039948 * A111559 A152406 A105623
Adjacent sequences: A111533 A111534 A111535 * A111537 A111538 A111539
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 06 2005
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