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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":"f899dd3cf819c813fe71f66f48eae5d902605ff4e75f1d07a43131b5fbd801d3","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-adb5eb667a229c4c","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 6).","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:           54,            20,             1;\n  1:          468,           324,            39,             1;\n  2:         4536,          5040,          1080,            64,            1;\n  3:        47952,         79056,         26460,          2700,           95,           1;\n  4:       536544,       1270080,        616896,         95760,         5670,         132,          1;\n  5:      6205248,      20948544,      14125104,       3111696,       274680,       10584,        175,         1;\n  6:     73063296,     353652480,     322392960,      96453504,     11866176,      671328,      18144,       224,       1;\n  7:    868361472,    6081889536,    7385411520,    2913900480,    478593360,    37204272,    1459080,     29160,     279,     1;\n  8:  10369949184,  106050954240,  170272751616,   86846411520,  18489889440,  1893253824,  100889712,   2898720,   44550,   340,   1;\n  9: 124137059328, 1867739710464, 3953243314944, 2571661737216, 694925256960, 91028095488, 6321324240, 244590192, 5363820, 65340, 407, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(72-24*x-5*x^2)*P(n-2,x) - (36*x-30*x^2)*d/dx P(n-2,x) + (18-4*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 54 + 20*x + x^2, P(1,x) = 468 + 324*x + 39*x^2 + x^3.","mathematica":"P[0, x_] := 54 + 20*x + x^2;\nP[1, x_] := 468 + 324*x + 39*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(72 - 24*x - 5*x^2)*P[n - 2, x] - (36*x - 30*x^2)*D[P[n - 2, x], x] + (18 - 4*x)*P[n - 1, x] + 6*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] = 54 + 20*x + x^2; v[2] = 468 + 324*x + 39*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(72 - 24*x - 5*x^2)*v[i - 2] - (36*x - 30*x^2)*deriv(v[i - 2], x) + (18 - 4*x)*v[i - 1] + 6*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-adb5eb667a229c4c","steps":["D+1"],"edges":["e-3cc6d9be2221b684"],"text":"A075501 \u2192 D+1"},"headline":"A075501 \u2192 D+1","chain":null}