A287314 Triangle read by rows, the coefficients of the polynomials generating the columns of A287316.

1, 0, 1, 0, -1, 2, 0, 4, -9, 6, 0, -33, 82, -72, 24, 0, 456, -1225, 1250, -600, 120, 0, -9460, 27041, -30600, 17700, -5400, 720, 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040, 0, -10643745, 33391954, -43471624, 31149496, -13524000, 3622080, -564480, 40320

Offset: 0

Author

Peter Luschny, May 27 2017

Keywords

• sign
• tabl

Comments

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

Examples

			Triangle starts:
[0] 1
[1] 0,      1
[2] 0,     -1,       2
[3] 0,      4,      -9,       6
[4] 0,    -33,      82,     -72,      24
[5] 0,    456,   -1225,    1250,    -600,    120
[6] 0,  -9460,   27041,  -30600,   17700,  -5400,    720
[7] 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040
...
For example let p4(x) = -33*x + 82*x^2 - 72*x^3 + 24*x^4 then p4(n) = A169712(n).
		

Crossrefs

Cf. A287316, A000384 (p2), A169711 (p3), A169712 (p4), A169713 (p5).

Cf. A000275(n), A212855.

Programs

Maple

A287314_row := proc(n) local k; sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): n!^2*coeff(%,z,n); seq(coeff(%,x,k), k=0..n) end:
for n from 0 to 8 do print(A287314_row(n)) od;
A287314_poly := proc(n) local k, x; sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): unapply(n!^2*coeff(%, z, n), x) end:
for n from 0 to 7 do A287314_poly(n) od;

Mathematica

nn = 10; e[x_] := Sum[x^n/n!^2, {n, 0, nn}];
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[
Series[e[x]^k, {x, 0, nn}], x], {k, 1, nn}]]], -1] // Grid (* Geoffrey Critzer, Jan 22 2021 *)

Formula

Sum_{k=0..n} abs(T(n,k)) = A000275(n) = A212855_row(2).