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b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"ddebea5de631e703ae7230cb5529215ef7288a47fcfce6a0528b6d2a16c9d182","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-7fdf5321df1ab451","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","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: 1,       7;\n   3: 1,      50,       139;\n   4: 1,     237,      2685,         4609;\n   5: 1,    1000,     30426,       172444,        214297;\n   6: 1,    4067,    277006,      3598426,      13152725,       12820999;\n   7: 1,   16350,   2260365,     57134324,     438656631,     1169007846,       937923283;\n   8: 1,   65497,  17359491,    777526955,   10784605475,    57018357003,    118943065321,      81109986433;\n   9: 1,  262100, 128655652,   9617733488,  220805932270,  2009414906468,   7984821768940,   13649288125736,    8094667157329;\n  10: 1, 1048527, 933090012, 111755676972, 4010001274566, 57831852719178, 382065425508756, 1206799379897076, 1745009081488521, 915646445959303;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(15*n-21)*x+(36*n-57)*x^2+(27*n-47)*x^3+(6*n-12)*x^4)*P(n-1,x) + (3*x+3*x^2-9*x^3-15*x^4-6*x^5)*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[((1 + (15*n - 21)*x + (36*n - 57)*x^2 + (27*n - 47)*x^3 + (6*n - 12)*x^4)*P[n - 1, x] + (3*x + 3*x^2 - 9*x^3 - 15*x^4 - 6*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\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] = ((1 + (15*n - 21)*x + (36*n - 57)*x^2 + (27*n - 47)*x^3 + (6*n - 12)*x^4)*v[i - 1] + (3*x + 3*x^2 - 9*x^3 - 15*x^4 - 6*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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-7fdf5321df1ab451","steps":["x \u21a6 x/(x+1)"],"edges":["e-7c43b0406d962969"],"text":"A290316 \u2192 x \u21a6 x/(x+1)"},"headline":"A290316 \u2192 x \u21a6 x/(x+1)","chain":null}