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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: 0, 2;\n  2: 0, 2,   3;\n  3: 0, 2,  12,     4;\n  4: 0, 2,  30,    40,     5;\n  5: 0, 2,  60,   200,   100,      6;\n  6: 0, 2, 105,   700,   875,    210,     7;\n  7: 0, 2, 168,  1960,  4900,   2940,   392,     8;\n  8: 0, 2, 252,  4704, 20580,  24696,  8232,   672,    9;\n  9: 0, 2, 360, 10080, 70560, 148176, 98784, 20160, 1080, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^4+5*n^3+8*n^2+4*n)*P(n,x) = (n^4+2*n^3-n^2-2*n+(10*n^4+20*n^3+8*n^2-2*n)*x)*P(n-1,x) + ((3*n^3+9*n^2+6*n)*x-(15*n^3+21*n^2)*x^2)*d/dx P(n-1,x) + ((6*n^2+10*n+2)*x^2+(6*n^2+10*n+2)*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[((n^4 + 2*n^3 - n^2 - 2*n + (10*n^4 + 20*n^3 + 8*n^2 - 2*n)*x)*P[n - 1, x] + ((3*n^3 + 9*n^2 + 6*n)*x - (15*n^3 + 21*n^2)*x^2)*D[P[n - 1, x], x] + ((6*n^2 + 10*n + 2)*x^2 + (6*n^2 + 10*n + 2)*x^3)*D[P[n - 1, x], {x, 2}])/(n^4 + 5*n^3 + 8*n^2 + 4*n)]];\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] = ((n^4 + 2*n^3 - n^2 - 2*n + (10*n^4 + 20*n^3 + 8*n^2 - 2*n)*x)*v[i - 1] + ((3*n^3 + 9*n^2 + 6*n)*x - (15*n^3 + 21*n^2)*x^2)*deriv(v[i - 1], x) + ((6*n^2 + 10*n + 2)*x^2 + (6*n^2 + 10*n + 2)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^4 + 5*n^3 + 8*n^2 + 4*n)); 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":["\u03b8+1"],"edges":["e-abb1e563f364e6b3"],"text":"A359363 \u2192 \u03b8+1"},"headline":"A359363 \u2192 \u03b8+1","chain":null}