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The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     1,      1;\n  2:     2,      3,      1;\n  3:     6,     12,      7,      1;\n  4:    22,     55,     46,     14,      1;\n  5:    92,    276,    300,    140,     25,      1;\n  6:   422,   1477,   1975,   1245,    365,     41,     1;\n  7:  2074,   8296,  13167,  10465,   4305,    847,    63,    1;\n  8: 10754,  48393,  88900,  85316,  45346,  12992,  1792,   92,   1;\n  9: 58202, 291010, 607308, 683172, 445998, 169134, 35196, 3516, 129, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n+6)*P(n,x) = (11*n^2+7*n-6+(10*n^2+5*n-3)*x)*P(n-1,x) - (12*n-12+(27*n-15)*x+(15*n-3)*x^2)*d/dx P(n-1,x) + (12+30*x+24*x^2+6*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((11*n^2 + 7*n - 6 + (10*n^2 + 5*n - 3)*x)*P[n - 1, x] - (12*n - 12 + (27*n - 15)*x + (15*n - 3)*x^2)*D[P[n - 1, x], x] + (12 + 30*x + 24*x^2 + 6*x^3)*D[P[n - 1, x], {x, 2}])/(n^2 + 5*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((11*n^2 + 7*n - 6 + (10*n^2 + 5*n - 3)*x)*v[i - 1] - (12*n - 12 + (27*n - 15)*x + (15*n - 3)*x^2)*deriv(v[i - 1], x) + (12 + 30*x + 24*x^2 + 6*x^3)*deriv(deriv(v[i - 1], x), x))/(n^2 + 5*n + 6)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-afb2f273dc5c03d1","steps":["x \u21a6 x+1"],"edges":["e-18ae8be6bae4e192"],"text":"A359363 \u2192 x \u21a6 x+1"},"headline":"A359363 \u2192 x \u21a6 x+1","chain":null}