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).
1, 1, 0, 1, 3, -1, 1, 17, -2, -1, 1, 80, 49, -27, 2, 1, 404, 733, -153, -49, 9, 1, 2359, 7860, 1622, -1606, 150, 9, 1, 16057, 80715, 58965, -17840, -3876, 1163, -50, 1, 125656, 858706, 1150722, 47365, -175756, 18239, 2359, -267, 1, 1112064, 9710898, 19571174, 7548463, -3175846, -491809, 194777, -9884, -413
Offset: 0
Examples
The triangle T(n, k) begins: n\k 0 1 2 3 4 5 6 7 8 9 ... 0: 1 1: 1 0 2: 1 3 -1 3: 1 17 -2 -1 4: 1 80 49 -27 2 5: 1 404 733 -153 -49 9 6: 1 2359 7860 1622 -1606 150 9 7: 1 16057 80715 58965 -17840 -3876 1163 -50 8: 1 125656 858706 1150722 47365 -175756 18239 2359 -267 9: 1 1112064 9710898 19571174 7548463 -3175846 -491809 194777 -9884 -413 ... 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.
Links
- Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Programs
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Mathematica
rows = 10; nmax = 30(*terms to find every gf*); T = Table[(-1)^(n - k) Sum[Binomial[-j - 1, -n - 1] StirlingS1[j, k], {j, 0, n}], {n, 0, nmax}, {k, 0, nmax}]; row[n_] := FindGeneratingFunction[Diagonal[T, -n], x] // Numerator // CoefficientList[-#, x]&; row[0] = {1}; row[1] = {1, 0}; Table[row[n], {n, 0, rows-1}] // Flatten (* Jean-François Alcover, Jan 26 2019 *)
Formula
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.
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