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A191936
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Triangle read by rows of Legendre-Stirling numbers of the first kind.
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3
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1, 1, 0, 1, -2, 0, 1, -8, 12, 0, 1, -20, 108, -144, 0, 1, -40, 508, -2304, 2880, 0, 1, -70, 1708, -17544, 72000, -86400, 0, 1, -112, 4648, -89280, 808848, -3110400, 3628800, 0, 1, -168, 10920, -349568, 5808528, -48405888, 177811200, -203212800, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,5
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COMMENTS
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LINKS
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FORMULA
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T(n, k) = ps(n-1, n-k), where ps(n, k) = ps(n-1, k-1) - n*(n-1)*ps(n-1, k), ps(n, 0) = 0, and ps(n, n) = 1. - G. C. Greubel, Jun 07 2021
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EXAMPLE
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Triangle begins:
1;
1, 0;
1, -2, 0;
1, -8, 12, 0;
1, -20, 108, -144, 0;
1, -40, 508, -2304, 2880, 0;
1, -70, 1708, -17544, 72000, -86400, 0;
1, -112, 4648, -89280, 808848, -3110400, 3628800, 0;
...
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MATHEMATICA
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ps[n_, k_]:= ps[n, k]= If[k==n, 1, If[k==0, 0, ps[n-1, k-1] - n*(n-1)*ps[n-1, k]]];
Table[ps[n-1, n-k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)
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PROG
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(Sage)
@CachedFunction
def ps(n, k):
if (k==n): return 1
elif (k==0): return 0
else: return ps(n-1, k-1) - n*(n-1)*ps(n-1, k)
flatten([[ps(n-1, n-k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jun 07 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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