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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":"1f3eae3965d7fc47b925f592c20dd7cbbadad36e46278190451950a3b00da2f7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"-1/2"}}}],"premises":[{"position":0,"node_id":"n-dd70177049267f47","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)"}]}],"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:    13,      37,      26;\n  1:    34,     146,     207,       97;\n  2:    89,     511,    1092,     1030,       362;\n  3:   233,    1675,    4787,     6801,      4805,      1351;\n  4:   610,    5268,   18858,    35826,     38105,     21519,      5042;\n  5:  1597,   16103,   69276,   164860,    234425,    199215,     93695,     18817;\n  6:  4181,   48209,  242236,   692874,   1234100,   1401790,    991760,    399628,     70226;\n  7: 10946,  142054,  816465,  2728051,   5840375,   8308845,   7855890,   4760502,   1677861,    262087;\n  8: 28657,  413375, 2674776, 10224458,  25571245,  43725171,  51773610,  41920128,  22214349,   6957625,   978122;\n  9: 75025, 1190807, 8566309, 36869443, 105498625, 210742847, 299907559, 304073697, 215268804, 101351960, 28562831, 3650401;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2+24*n+18+(16*n^2+48*n+36)*x+(8*n^2+24*n+18)*x^2)*P(n,x) = (4*n^2-34*n-33+(20*n^2-186*n-189)*x+(64*n^2-342*n-387)*x^2+(120*n^2-226*n-339)*x^3+(108*n^2-108)*x^4+(36*n^2+36*n)*x^5)*P(n-1,x) - (10*n-57+(92*n-330)*x+(336*n-756)*x^2+(628*n-858)*x^3+(638*n-483)*x^4+(336*n-108)*x^5+72*n*x^6)*d/dx P(n-1,x) + (15+123*x+426*x^2+810*x^3+915*x^4+615*x^5+228*x^6+36*x^7)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 13 + 37*x + 26*x^2.","mathematica":"P[0, x_] := 13 + 37*x + 26*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 34*n - 33 + (20*n^2 - 186*n - 189)*x + (64*n^2 - 342*n - 387)*x^2 + (120*n^2 - 226*n - 339)*x^3 + (108*n^2 - 108)*x^4 + (36*n^2 + 36*n)*x^5)*P[n - 1, x] - (10*n - 57 + (92*n - 330)*x + (336*n - 756)*x^2 + (628*n - 858)*x^3 + (638*n - 483)*x^4 + (336*n - 108)*x^5 + 72*n*x^6)*D[P[n - 1, x], x] + (15 + 123*x + 426*x^2 + 810*x^3 + 915*x^4 + 615*x^5 + 228*x^6 + 36*x^7)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18 + (16*n^2 + 48*n + 36)*x + (8*n^2 + 24*n + 18)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 37*x + 26*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 - 34*n - 33 + (20*n^2 - 186*n - 189)*x + (64*n^2 - 342*n - 387)*x^2 + (120*n^2 - 226*n - 339)*x^3 + (108*n^2 - 108)*x^4 + (36*n^2 + 36*n)*x^5)*v[i - 1] - (10*n - 57 + (92*n - 330)*x + (336*n - 756)*x^2 + (628*n - 858)*x^3 + (638*n - 483)*x^4 + (336*n - 108)*x^5 + 72*n*x^6)*deriv(v[i - 1], x) + (15 + 123*x + 426*x^2 + 810*x^3 + 915*x^4 + 615*x^5 + 228*x^6 + 36*x^7)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18 + (16*n^2 + 48*n + 36)*x + (8*n^2 + 24*n + 18)*x^2)); 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-dfab35176bd08e72","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-23e971f1ed1ce0e7","e-699dac091acaad0c"],"text":"A207606 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A207606 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}