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A111823 Matrix log of triangle A111820, which shifts columns left and up under matrix 5th power; these terms are the result of multiplying each element in row n and column k by (n-k)!. 8
0, 1, 0, -3, 5, 0, 16, -15, 25, 0, 2814, 80, -75, 125, 0, -1092180, 14070, 400, -375, 625, 0, -3603928080, -5460900, 70350, 2000, -1875, 3125, 0, 58978973128440, -18019640400, -27304500, 351750, 10000, -9375, 15625, 0, 5974833380453777520 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Column k equals 5^k multiplied by column 0 (A111824) when ignoring zeros above the diagonal.
LINKS
FORMULA
T(n, k) = 5^k*T(n-k, 0) = A111824(n-k) for n>=k>=0.
EXAMPLE
Matrix log of A111820, with factorial denominators, begins:
0;
1/1!, 0;
-3/2!, 5/1!, 0;
16/3!, -15/2!, 25/1!, 0;
2814/4!, 80/3!, -75/2!, 125/1!, 0;
-1092180/5!, 14070/4!, 400/3!, -375/2!, 625/1!, 0; ...
PROG
(PARI) T(n, k, q=5)=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]))
CROSSREFS
Cf. A111820, A111824 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111828 (q=6), A111833 (q=7), A111838 (q=8).
Sequence in context: A349605 A318204 A306637 * A113039 A093016 A031018
KEYWORD
frac,sign,tabl
AUTHOR
STATUS
approved

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Last modified June 14 05:27 EDT 2024. Contains 373393 sequences. (Running on oeis4.)