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Search: id:A111815
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| A111815 |
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Matrix log of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!. |
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+0
9
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| 0, 1, 0, -1, 3, 0, -3, -3, 9, 0, 150, -9, -9, 27, 0, 1236, 450, -27, -27, 81, 0, -2555748, 3708, 1350, -81, -81, 243, 0, -64342116, -7667244, 11124, 4050, -243, -243, 729, 0, 5885700899760, -193026348, -23001732, 33372, 12150, -729, -729, 2187, 0
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Column k equals 3^k multiplied by column 0 (A111816) when ignoring zeros above the diagonal.
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FORMULA
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T(n, k) = 3^k*T(n-k, 0) = A111816(n-k) for n>=k>=0.
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EXAMPLE
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Matrix log of A078122, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, 3/1!, 0;
-3/3!, -3/2!, 9/1!, 0;
150/4!, -9/3!, -9/2!, 27/1!, 0;
1236/5!, 450/4!, -27/3!, -27/2!, 81/1!, 0;
-2555748/6!, 3708/5!, 1350/4!, -81/3!, -81/2!, 243/1!, 0; ...
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PROGRAM
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(PARI) {T(n, k, q=3)=local(A=Mat(1), B); if(n<k|k<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i|j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); B=sum(i=1, #A, -(A^0-A)^i/i); return((n-k)!*B[n+1, k+1]))}
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CROSSREFS
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Cf. A078122, A111816 (column 0), A111840 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).
Sequence in context: A066851 A167223 A078907 this_sequence A127753 A073367 A111862
Adjacent sequences: A111812 A111813 A111814 this_sequence A111816 A111817 A111818
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KEYWORD
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frac,sign,tabl
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AUTHOR
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Gottfried Helms (helms(AT)uni-kassel.de) and Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2005
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