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for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"1d7fb9d0b2bd69e2d2f3fce7107627105886fca365f82207ee0cf1aea88377fe","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-deb68900633836aa","role":"sequence","name":"basis_map_A008277 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         1;\n   2:         3,          5;\n   3:        15,         50,          39;\n   4:       105,        525,         819,          407;\n   5:       945,       6300,       14742,        14652,         5281;\n   6:     10395,      86625,      270270,       402930,       290455,        81597;\n   7:    135135,    1351350,     5270265,     10476180,     11327745,      6364566,      1459015;\n   8:   2027025,   23648625,   110675565,    274999725,    396471075,    334139715,    153196575,     29582303;\n   9:  34459425,  459459000,  2508646140,   7479992520,  13480016550,  15147667080,  10417367100,   4023193208,    669822849;\n  10: 654729075, 9820936125, 61282641420, 213179786820, 461016566010, 647562767670, 593789924700, 343983019284, 114539707179, 16737058997;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(2*n+4)*x-(2*n-9)*x^2-(2*n-4)*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (2*n + 4)*x - (2*n - 9)*x^2 - (2*n - 4)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 1 + (2*n + 4)*x - (2*n - 9)*x^2 - (2*n - 4)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A008277","x \u21a6 x/(x+1)"],"edges":["e-0664c5d711618d77","e-884e9cc49f200fc3"],"text":"A161198 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)"},"headline":"A161198 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)","chain":null}
