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A379149 Specialization of the Elementary Symmetric Functions e(n) at x_i -> Euler phi(i).

%I A379149 #25 Dec 22 2024 10:36:16
%S A379149 1,1,1,1,2,1,1,4,5,2,1,6,13,12,4,1,10,37,64,52,16,1,12,57,138,180,120,
%T A379149 32,1,18,129,480,1008,1200,752,192,1,22,201,996,2928,5232,5552,3200,
%U A379149 768,1,28,333,2202,8904,22800,36944,36512,19968,4608,1,32,445,3534,17712,58416,128144,184288,166016,84480,18432
%N A379149 Specialization of the Elementary Symmetric Functions e(n) at x_i -> Euler phi(i).
%C A379149 Triangular table with alternating signed sum equal to 0 for n>0,
%C A379149   1
%C A379149   1,-1
%C A379149   1,-2,1
%C A379149   1,-4,5,-2
%C A379149   1,-6,13,-12,4
%C A379149   ..
%C A379149 and with alternating signed weighted sum (first moment) also equal to 0 for n>1,
%C A379149   0
%C A379149   0,-1
%C A379149   0,-2,2
%C A379149   0,-4,10,-6
%C A379149   0,-6,26,-36,16
%C A379149   ..
%C A379149 also when shifting the weights to start at 1,
%C A379149   1
%C A379149   1,-2
%C A379149   1,-4,3
%C A379149   1,-8,15,-8
%C A379149   1,-12,39,-48,20
%H A379149 Alois P. Heinz, <a href="/A379149/b379149.txt">Rows n = 0..140, flattened</a>
%H A379149 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/801399">Specializations of elementary symmetric polynomials</a>
%F A379149 T(n,k) = [x^k] Product_{j=1..n} (1 + x*phi(j)). - _Andrew Howroyd_, Dec 16 2024
%e A379149 Triangle begins:
%e A379149   1;
%e A379149   1,  1;
%e A379149   1,  2,  1;
%e A379149   1,  4,  5,   2;
%e A379149   1,  6, 13,  12,   4;
%e A379149   1, 10, 37,  64,  52,  16;
%e A379149   1, 12, 57, 138, 180, 120, 32;
%e A379149   ...
%p A379149 b:= proc(n) option remember; `if`(n=0, 1,
%p A379149        b(n-1)*(1+x*numtheory[phi](n)))
%p A379149     end:
%p A379149 T:= (n, k)-> coeff(b(n), x, k):
%p A379149 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Dec 16 2024
%t A379149 Table[CoefficientList[Expand@Product[z EulerPhi[k]+1,{k,0,n}],z,n+1],{n,0,10}]
%o A379149 (PARI) row(n) = Vecrev(prod(k=1, n, 1 + 'x * eulerphi(k))) \\ _Andrew Howroyd_, Dec 16 2024
%Y A379149 Columns k=0-1 give: A000012, A002088.
%Y A379149 Main diagonal gives A001088.
%Y A379149 T(n,n-1) gives A067578.
%Y A379149 Cf. A000010.
%K A379149 nonn,tabl
%O A379149 0,5
%A A379149 _Wouter Meeussen_, Dec 16 2024