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%I #17 Jan 25 2021 20:15:50
%S 1,0,1,0,-1,2,0,4,-9,6,0,-33,82,-72,24,0,456,-1225,1250,-600,120,0,
%T -9460,27041,-30600,17700,-5400,720,0,274800,-826336,1011017,-661500,
%U 249900,-52920,5040,0,-10643745,33391954,-43471624,31149496,-13524000,3622080,-564480,40320
%N Triangle read by rows, the coefficients of the polynomials generating the columns of A287316.
%C The zeta polynomials for the poset P_n of ordered pairs (S,T) where S,T are subsets of [n] with |S| = |T| ordered component-wise by inclusion. - _Geoffrey Critzer_, Jan 22 2021
%F Sum_{k=0..n} abs(T(n,k)) = A000275(n) = A212855_row(2).
%e Triangle starts:
%e [0] 1
%e [1] 0, 1
%e [2] 0, -1, 2
%e [3] 0, 4, -9, 6
%e [4] 0, -33, 82, -72, 24
%e [5] 0, 456, -1225, 1250, -600, 120
%e [6] 0, -9460, 27041, -30600, 17700, -5400, 720
%e [7] 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040
%e ...
%e For example let p4(x) = -33*x + 82*x^2 - 72*x^3 + 24*x^4 then p4(n) = A169712(n).
%p A287314_row := proc(n) local k; sum(z^k/k!^2, k = 0..infinity);
%p series(%^x, z=0, n+1): n!^2*coeff(%,z,n); seq(coeff(%,x,k), k=0..n) end:
%p for n from 0 to 8 do print(A287314_row(n)) od;
%p A287314_poly := proc(n) local k, x; sum(z^k/k!^2, k = 0..infinity);
%p series(%^x, z=0, n+1): unapply(n!^2*coeff(%, z, n), x) end:
%p for n from 0 to 7 do A287314_poly(n) od;
%t nn = 10; e[x_] := Sum[x^n/n!^2, {n, 0, nn}];
%t f[list_] := CoefficientList[InterpolatingPolynomial[Table[{i, list[[i]]}, {i, 1, nn}], m], m];Drop[Map[f,Transpose[Table[Table[n!^2, {n, 0, nn}] CoefficientList[
%t Series[e[x]^k, {x, 0, nn}], x], {k, 1, nn}]]], -1] // Grid (* _Geoffrey Critzer_, Jan 22 2021 *)
%Y Cf. A287316, A000384 (p2), A169711 (p3), A169712 (p4), A169713 (p5).
%Y Cf. A000275(n), A212855.
%K sign,tabl
%O 0,6
%A _Peter Luschny_, May 27 2017
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