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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":"0bf06fc7627898fbc65f6b519dccf2a76b1b777a79b13fd4b70f584aae7d60b6","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-aac530d688cc3150","role":"sequence","name":"Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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:       0;\n  1:       1;\n  2:      17,          2;\n  3:     118,        236,           3;\n  4:     729,       5362,        2187,            4;\n  5:    4400,      83490,      125235,        17600,            5;\n  6:   26431,    1110480,     4047735,      2220960,       132155,           6;\n  7:  158622,   13633692,   100370055,    133826740,     34084230,      951732,          7;\n  8:  951773,  160069486,  2145963468,   5647707980,   3576605780,   480208458,    6662411,        8;\n  9: 5710684, 1830878222, 41880899277, 193664401504, 242080501880, 83761798554, 6408073777, 45685472, 9;\n","recurrence_sentence":"The row polynomials satisfy (5*n-4)*P(n,x) = -(125*n+25)*x^2*P(n-4,x) - ((50*n+10)*x+(250*n^2-650*n-15)*x^2)*P(n-3,x) - ((125*n+25)*x^2-(125*n+25)*x^3)*d/dx P(n-3,x) - (30*n+6+(50*n^2-180*n+12)*x+(125*n^3-550*n^2+640*n-224)*x^2)*P(n-2,x) - ((25*n+5)*x+(125*n^2-225*n+75)*x^2-(125*n^2-200*n+80)*x^3)*d/dx P(n-2,x) + (35*n-23+(50*n^2-130*n+72)*x)*P(n-1,x) + ((25*n-20)*x-(25*n-20)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 17 + 2*x, P(3,x) = 118 + 236*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 17 + 2*x;\nP[3, x_] := 118 + 236*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(125*n + 25)*x^2*P[n - 4, x] - ((50*n + 10)*x + (250*n^2 - 650*n - 15)*x^2)*P[n - 3, x] - ((125*n + 25)*x^2 - (125*n + 25)*x^3)*D[P[n - 3, x], x] - (30*n + 6 + (50*n^2 - 180*n + 12)*x + (125*n^3 - 550*n^2 + 640*n - 224)*x^2)*P[n - 2, x] - ((25*n + 5)*x + (125*n^2 - 225*n + 75)*x^2 - (125*n^2 - 200*n + 80)*x^3)*D[P[n - 2, x], x] + (35*n - 23 + (50*n^2 - 130*n + 72)*x)*P[n - 1, x] + ((25*n - 20)*x - (25*n - 20)*x^2)*D[P[n - 1, x], x])/(5*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 17 + 2*x; v[4] = 118 + 236*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(125*n + 25)*x^2*v[i - 4] - ((50*n + 10)*x + (250*n^2 - 650*n - 15)*x^2)*v[i - 3] - ((125*n + 25)*x^2 - (125*n + 25)*x^3)*deriv(v[i - 3], x) - (30*n + 6 + (50*n^2 - 180*n + 12)*x + (125*n^3 - 550*n^2 + 640*n - 224)*x^2)*v[i - 2] - ((25*n + 5)*x + (125*n^2 - 225*n + 75)*x^2 - (125*n^2 - 200*n + 80)*x^3)*deriv(v[i - 2], x) + (35*n - 23 + (50*n^2 - 130*n + 72)*x)*v[i - 1] + ((25*n - 20)*x - (25*n - 20)*x^2)*deriv(v[i - 1], x))/(5*n - 4)); 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-aac530d688cc3150","steps":["D"],"edges":["e-db796833c5f87fc1"],"text":"A144442 \u2192 D"},"headline":"A144442 \u2192 D","chain":null}