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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":"fa8e710b278bdbe97f4f64149554cf8cc55f4fe107c184c383a2c90d7ce238ff","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-a06f8ee83f308dcc","role":"sequence","name":"basis_map_A013610 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,             7;\n  2:          32,           100,            77;\n  3:         384,          1680,          2424,           1155;\n  4:        6144,         34176,         70704,          64536,          21945;\n  5:      122880,        824832,       2200128,        2916912,        1923228,         504735;\n  6:     2949120,      23113728,      75073536,      129409344,      124914096,       64040796,       13627845;\n  7:    82575360,     738607104,    2818584576,     5950741248,     7509284352,     5665479264,     2366844336,      422463195;\n  8:  2642411520,   26525564928,  116045955072,   289074180096,   448573042944,   444120288768,   274030792992,    96358374000,   14786211825;\n  9: 95126814720, 1057974386688, 5212151414784, 14932462731264, 27422522569728, 33482679070464, 27185799809664, 14156102390688, 4290280211700, 576662261175;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (4*n+(8*n+3)*x+(4*n+3)*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[((4*n + (8*n + 3)*x + (4*n + 3)*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] = ((4*n + (8*n + 3)*x + (4*n + 3)*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-9c5c208df8605708","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-bf3b7a758db4c427","e-54593ea8bd5c8a5d"],"text":"A225471 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A225471 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}