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A111544
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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.
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10
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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
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OFFSET
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0,4
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COMMENTS
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Column 0 equals A111530 (related to log of factorial series). Column 3 (A111547) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111549.
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LINKS
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FORMULA
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T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+2, 2)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+3, 2) = 3*T(n+1, 0), T(n+4, 4) = T(n+1, 0), for n>=0.
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EXAMPLE
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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.
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[n<k || k<0, 0, If[n == k, 1, If[n == k + 1, n, k T[n, k + 1] + Sum[T[j + 2, 2] T[n, j + k + 1], {j, 0, n - k - 1}]]]];
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PROG
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(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+2, 2)*T(n, j+k+1)))))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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