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A158198 Triangle T(n, k) = Sum_{j=0..k-1} (-1)^j*binomial(k, j+1)*(k-j)^(n-k), read by rows.

%I A158198 #12 Sep 08 2022 08:45:42
%S A158198 1,1,1,1,3,1,1,7,4,1,1,15,16,5,1,1,31,58,25,6,1,1,63,196,125,36,7,1,1,
%T A158198 127,634,601,216,49,8,1,1,255,1996,2765,1296,343,64,9,1,1,511,6178,
%U A158198 12265,7656,2401,512,81,10,1,1,1023,18916,52925,44136,16807,4096,729,100,11,1
%N A158198 Triangle T(n, k) = Sum_{j=0..k-1} (-1)^j*binomial(k, j+1)*(k-j)^(n-k), read by rows.
%H A158198 G. C. Greubel, <a href="/A158198/b158198.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A158198 T(n, k) = Sum_{j=0..k-1} (-1)^j*binomial(k, j+1)*(k-j)^(n-k).
%e A158198 Triangle begins
%e A158198   1;
%e A158198   1,   1;
%e A158198   1,   3,    1;
%e A158198   1,   7,    4,    1;
%e A158198   1,  15,   16,    5,    1;
%e A158198   1,  31,   58,   25,    6,   1;
%e A158198   1,  63,  196,  125,   36,   7,  1;
%e A158198   1, 127,  634,  601,  216,  49,  8, 1;
%e A158198   1, 255, 1996, 2765, 1296, 343, 64, 9, 1;
%e A158198 a(8,4) = 1*4*4^4 - 1*6*3^4 + 1*4*2^4 - 1*1*1^4 = 1024 - 486 + 64 - 1 = 601.
%t A158198 Table[Sum[(-1)^j*Binomial[k, j+1]*(k-j)^(n-k), {j, 0, k-1}], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jun 26 2021 *)_
%o A158198 (PARI) tabl(nn) = {for (n=1, nn, for (k=1, n, print1(sum(i=0, k-1, (-1)^i*binomial(k,i+1)*(k-i)^(n-k)), ", ");); print(););} \\ _Michel Marcus_, Jun 19 2013
%o A158198 (Magma)
%o A158198 [(&+[(-1)^j*Binomial(k, j+1)*(k-j)^(n-k): j in [0..k-1]]): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jun 26 2021
%o A158198 (Sage)
%o A158198 flatten([[ sum( (-1)^j*binomial(k, j+1)*(k-j)^(n-k) for j in (0..k-1)) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jun 26 2021
%Y A158198 Column 2 is A000225, column 3 is A168583. - _Michel Marcus_, Jun 19 2013
%K A158198 nonn,tabl
%O A158198 1,5
%A A158198 Thomas J Engelsma (tom(AT)opertech.com), Mar 13 2009