%I #13 Jan 26 2019 14:27:59

%S 1,1,0,1,3,-1,1,17,-2,-1,1,80,49,-27,2,1,404,733,-153,-49,9,1,2359,

%T 7860,1622,-1606,150,9,1,16057,80715,58965,-17840,-3876,1163,-50,1,

%U 125656,858706,1150722,47365,-175756,18239,2359,-267,1,1112064,9710898,19571174,7548463,-3175846,-491809,194777,-9884,-413

%N Triangle T(n, k) read by rows: row n gives the coefficients of the row polynomials of the (n+1)-th diagonal sequence of the Sheffer triangle A094816 (special Poisson-Charlier).

%C The o.g.f. of the (n+1)-th diagonal sequence of the Sheffer triangle (e^x, -(log(1-x))) (the product of two Sheffer triangles A007318*A132393 = Pascal*|Stirling1|) is P(n, x)/(1 - x)^{2*n+1}, for n >= 0., with the numerator polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k.

%C O.g.f.'s for diagonal sequences of Sheffer matrices (lower triangular) can be computed via Lagrange's theorem. For the special case of Jabotinsky matrices (1, f(x)) this has been done by P. Bala (see the link under A112007), and the method can be generalized to Sheffer (g(x), f(x)), as shown in the W. Lang link given below.

%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1708.01421">On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.

%F T(n, k) = [x^n] P(n, x) with the numerator polynomials (in rising powers) of the o.g.f. of the (n+1)-th diagonal sequence of the triangle A094816. See the comment above.

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 ...

%e 0: 1

%e 1: 1 0

%e 2: 1 3 -1

%e 3: 1 17 -2 -1

%e 4: 1 80 49 -27 2

%e 5: 1 404 733 -153 -49 9

%e 6: 1 2359 7860 1622 -1606 150 9

%e 7: 1 16057 80715 58965 -17840 -3876 1163 -50

%e 8: 1 125656 858706 1150722 47365 -175756 18239 2359 -267

%e 9: 1 1112064 9710898 19571174 7548463 -3175846 -491809 194777 -9884 -413

%e ...

%e n = 2: the o.g.f. of the third diagonal of triangle A094816, [1, 8, 29, 75, 160, ...] = A290312 is (1 + 3*x - x^2)/(1 - x)^5.

%t rows = 10; nmax = 30(*terms to find every gf*);

%t T = Table[(-1)^(n - k) Sum[Binomial[-j - 1, -n - 1] StirlingS1[j, k], {j, 0, n}], {n, 0, nmax}, {k, 0, nmax}];

%t row[n_] := FindGeneratingFunction[Diagonal[T, -n], x] // Numerator // CoefficientList[-#, x]&; row[0] = {1}; row[1] = {1, 0};

%t Table[row[n], {n, 0, rows-1}] // Flatten (* _Jean-François Alcover_, Jan 26 2019 *)

%Y Cf. A094816, A290312, A290313, A290314.

%K sign,tabl

%O 0,5

%A _Wolfdieter Lang_, Jul 28 2017