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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"acbec7308e62643f14df52dc1a0009ca9de0baad2c3d848b27fb771850d6b127","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-fc18ad0ea0731402","role":"sequence","name":"Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).","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:           70,            23,              1;\n  1:          686,           427,             45,             1;\n  2:         7546,          7595,           1435,            74,             1;\n  3:        91238,        136171,          40425,          3605,           110,            1;\n  4:      1176490,       2504243,        1082851,        147490,          7595,          153,           1;\n  5:     15764966,      47412547,       28487865,       5524701,        425320,        14210,         203,         1;\n  6:    215768266,     921779915,      747659395,     197347794,      21227241,      1043406,       24402,       260,       1;\n  7:   2986166918,   18313125691,    19720913625,    6871289845,     988755810,     66908667,     2274090,     39270,     324,     1;\n  8:  41564215210,  369886926563,   524396770051,  236141543330,   44110727815,   3938434731,   182156667,   4527600,   60060,   395,   1;\n  9: 580204161446, 7561499233267, 14067180928185, 8069162433101, 1914626884940, 218994201580, 13217411361, 442946427, 8392230, 88165, 473, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(98-35*x-6*x^2)*P(n-2,x) - (49*x-42*x^2)*d/dx P(n-2,x) + (21-5*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 70 + 23*x + x^2, P(1,x) = 686 + 427*x + 45*x^2 + x^3.","mathematica":"P[0, x_] := 70 + 23*x + x^2;\nP[1, x_] := 686 + 427*x + 45*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(98 - 35*x - 6*x^2)*P[n - 2, x] - (49*x - 42*x^2)*D[P[n - 2, x], x] + (21 - 5*x)*P[n - 1, x] + 7*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 70 + 23*x + x^2; v[2] = 686 + 427*x + 45*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(98 - 35*x - 6*x^2)*v[i - 2] - (49*x - 42*x^2)*deriv(v[i - 2], x) + (21 - 5*x)*v[i - 1] + 7*x*deriv(v[i - 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-fc18ad0ea0731402","steps":["D+1"],"edges":["e-179a4951b5284356"],"text":"A075502 \u2192 D+1"},"headline":"A075502 \u2192 D+1","chain":null}