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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":"11c6ad28f6fee684f8eda64097bb2f460a19845cecc9893b0fd73f5e4c537ab7","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-5c56442adc0376c2","role":"sequence","name":"Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 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:            5,             6;\n   3:           45,           105,            61;\n   4:          585,          2010,          2295,            871;\n   5:         9945,         44955,         76020,          57005,          15996;\n   6:       208845,       1168020,       2607780,        2905695,        1615980,         358891;\n   7:      5221125,      34754265,      96230295,      141881880,      117493845,       51818760,        9509641;\n   8:    151412625,    1167934950,    3855423285,     7060897095,     7748832105,     5095995240,     1859687585,      290528316;\n   9:   4996616625,   43797929175,  167750040150,   366702807555,   500439275595,   436613354415,   237833143380,    73956957975,   10051973371;\n  10: 184874815125, 1814219807400, 7903812366450, 20065028163075, 32712811130475, 35520939320265, 25689602239560, 11933258109615, 3230778665835, 388433817091;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (4*n-3+(7*n-3)*x+(2*n+2)*x^2-(n-2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[((4*n - 3 + (7*n - 3)*x + (2*n + 2)*x^2 - (n - 2)*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, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n - 3 + (7*n - 3)*x + (2*n + 2)*x^2 - (n - 2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5c56442adc0376c2","steps":["x \u21a6 x/(x+1)"],"edges":["e-d4e82a1ebb564a7a"],"text":"A049029 \u2192 x \u21a6 x/(x+1)"},"headline":"A049029 \u2192 x \u21a6 x/(x+1)","chain":null}