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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":"a3f47c6373d2b183a8ccc8e0ee43984929aa89a2eca6ed9b3fd3061e5140b1a0","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-f4c6c763b9527d17","role":"sequence","name":"basis_map_A008277 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:           4,            5;\n  2:          32,           77,            46;\n  3:         384,         1353,          1581,            613;\n  4:        6144,        28377,         48954,          37399,          10679;\n  5:      122880,       700365,       1591575,        1803080,        1018565,         229576;\n  6:     2949120,     19966725,      56174580,       84079500,       70625175,       31571364,        5868715;\n  7:    82575360,    646864785,    2166710805,     4023344535,     4473612780,     2978996937,     1100140972,     173833661;\n  8:  2642411520,  23493497265,   91202353830,   201937194165,   278963531175,   246232343049,   135628408328,   42626875841,    5853205468;\n  9: 95126814720, 945889332885, 4172848568595, 10720773767970, 17679128919795, 19407490747659, 14183600403987, 6655023373464, 1819239885099, 220767370219;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (4*n+(7*n+2)*x+(2*n+3)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx 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[((4*n + (7*n + 2)*x + (2*n + 3)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], 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] = ((4*n + (7*n + 2)*x + (2*n + 3)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A008277","x \u21a6 x/(x+1)"],"edges":["e-3a6a7b6037078bc1","e-96aea6882beef27d"],"text":"A225471 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)"},"headline":"A225471 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)","chain":null}