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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":"a81feb0ab86015a42498134e8f4dc1900da3f13f03c1f4e3ee91253ccc83fbb0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-3255c4dcfe77aa0b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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,           3;\n  2:          2,          10,           12;\n  3:         10,          64,          130,            84;\n  4:        100,         760,         2068,          2400,          1008;\n  5:       1700,       14820,        49596,         80092,         62736,         19152;\n  6:      44200,      432920,      1704456,       3471080,       3873712,       2254560,        536256;\n  7:    1635400,    17741840,     79948752,     194903744,     278699464,     234493488,     107769312,      20913984;\n  8:   81770000,   972132800,   4920013280,   13902522304,   24069967888,   26217046528,   17582126976,    6649703424,   1087527168;\n  9: 5315050000, 68667222000, 384933760800, 1233304839520, 2496016907088, 3316795872816, 2899380370816, 1610233229952, 516219395328, 72864320256;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n^2-2*n+2+(2*n^2-4*n+6)*x+(n^2-2*n+4)*x^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^2 - 2*n + 2 + (2*n^2 - 4*n + 6)*x + (n^2 - 2*n + 4)*x^2)*P[n - 1, x])/(1 + x)]];\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^2 - 2*n + 2 + (2*n^2 - 4*n + 6)*x + (n^2 - 2*n + 4)*x^2)*v[i - 1])/(1 + x)); 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-54284d24a094e2d3","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-327e7f94e2e44f03","e-9c1108f30a487478"],"text":"A269944 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A269944 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}