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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"cf55691c274614c3e9d4b51fbd0a4002f45dc91dfe68f6a400149500e634c87d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a6a633b989da1879","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","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:        12,          5;\n  2:       119,        120,         25;\n  3:      1078,       1785,        900,        125;\n  4:      9261,      21560,      17850,       6000,        625;\n  5:     76832,     231525,     269500,     148750,      37500,       3125;\n  6:    621859,    2304960,    3472875,    2695000,    1115625,     225000,      15625;\n  7:   4941258,   21765065,   40336800,   40516875,   23581250,    7809375,    1312500,     78125;\n  8:  38706521,  197650320,  435301300,  537824000,  405168750,  188650000,   52062500,   7500000,   390625;\n  9: 299769652, 1741793445, 4447132200, 6529519500, 6050520000, 3646518750, 1414875000, 334687500, 42187500, 1953125;\n","recurrence_sentence":"The row polynomials satisfy (5*n+2+5*x)*P(n,x) = (35*n+49+(25*n+70)*x+25*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[((35*n + 49 + (25*n + 70)*x + 25*x^2)*P[n - 1, x])/(5*n + 2 + 5*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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