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A290311 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).

%I A290311 #13 Jan 26 2019 14:27:59
%S A290311 1,1,0,1,3,-1,1,17,-2,-1,1,80,49,-27,2,1,404,733,-153,-49,9,1,2359,
%T A290311 7860,1622,-1606,150,9,1,16057,80715,58965,-17840,-3876,1163,-50,1,
%U A290311 125656,858706,1150722,47365,-175756,18239,2359,-267,1,1112064,9710898,19571174,7548463,-3175846,-491809,194777,-9884,-413
%N A290311 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 A290311 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 A290311 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 A290311 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 A290311 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 A290311 The triangle T(n, k) begins:
%e A290311 n\k 0       1       2        3       4        5       6      7     8    9 ...
%e A290311 0:  1
%e A290311 1:  1       0
%e A290311 2:  1       3      -1
%e A290311 3:  1      17      -2       -1
%e A290311 4:  1      80      49      -27       2
%e A290311 5:  1     404     733     -153     -49        9
%e A290311 6:  1    2359    7860     1622   -1606      150       9
%e A290311 7:  1   16057   80715    58965  -17840    -3876    1163    -50
%e A290311 8:  1  125656  858706  1150722   47365  -175756   18239   2359  -267
%e A290311 9:  1 1112064 9710898 19571174 7548463 -3175846 -491809 194777 -9884 -413
%e A290311 ...
%e A290311 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 A290311 rows = 10; nmax = 30(*terms to find every gf*);
%t A290311 T = Table[(-1)^(n - k) Sum[Binomial[-j - 1, -n - 1] StirlingS1[j, k], {j, 0, n}], {n, 0, nmax}, {k, 0, nmax}];
%t A290311 row[n_] := FindGeneratingFunction[Diagonal[T, -n], x] // Numerator // CoefficientList[-#, x]&; row[0] = {1}; row[1] = {1, 0};
%t A290311 Table[row[n], {n, 0, rows-1}] // Flatten (* _Jean-François Alcover_, Jan 26 2019 *)
%Y A290311 Cf. A094816, A290312, A290313, A290314.
%K A290311 sign,tabl
%O A290311 0,5
%A A290311 _Wolfdieter Lang_, Jul 28 2017