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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":"69e5bd36c92de6a7d1dab02e7251bdd7c62b40dd66cc198eda1795e343333370","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-817c504263a54deb","role":"sequence","name":"reversal of Coefficient triangle of certain polynomials N(5; m,x).","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:     7,      1;\n  2:    35,     16,      1;\n  3:   140,    132,     27,       1;\n  4:   462,    768,    324,      40,      1;\n  5:  1302,   3450,   2550,     650,     55,      1;\n  6:  3234,  12672,  14850,    6600,   1155,     72,     1;\n  7:  7260,  39732,  68607,   48125,  14630,   1890,    91,    1;\n  8: 15015, 109824, 264264,  272272, 130130,  29120,  2912,  112,   1;\n  9: 29029, 274131, 880308, 1261260, 882882, 309582, 53508, 4284, 135, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^4+3*n^3-3*n^2+35*n+(2*n^3+6*n^2-20*n)*x+(n^2+5*n)*x^2)*P(n,x) = (n^4+10*n^3+31*n^2+60*n+150+(3*n^4+6*n^3+18*n^2+50*n-125)*x+(6*n^3+8*n^2-24*n+40)*x^2+(3*n^2+8*n-5)*x^3)*P(n-1,x) + ((2*n^3+4*n^2+10*n+25)*x-(2*n^3+4*n+35)*x^2-(4*n^2+4*n-15)*x^3-(2*n+5)*x^4)*d/dx 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[((n^4 + 10*n^3 + 31*n^2 + 60*n + 150 + (3*n^4 + 6*n^3 + 18*n^2 + 50*n - 125)*x + (6*n^3 + 8*n^2 - 24*n + 40)*x^2 + (3*n^2 + 8*n - 5)*x^3)*P[n - 1, x] + ((2*n^3 + 4*n^2 + 10*n + 25)*x - (2*n^3 + 4*n + 35)*x^2 - (4*n^2 + 4*n - 15)*x^3 - (2*n + 5)*x^4)*D[P[n - 1, x], x])/(n^4 + 3*n^3 - 3*n^2 + 35*n + (2*n^3 + 6*n^2 - 20*n)*x + (n^2 + 5*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^4 + 10*n^3 + 31*n^2 + 60*n + 150 + (3*n^4 + 6*n^3 + 18*n^2 + 50*n - 125)*x + (6*n^3 + 8*n^2 - 24*n + 40)*x^2 + (3*n^2 + 8*n - 5)*x^3)*v[i - 1] + ((2*n^3 + 4*n^2 + 10*n + 25)*x - (2*n^3 + 4*n + 35)*x^2 - (4*n^2 + 4*n - 15)*x^3 - (2*n + 5)*x^4)*deriv(v[i - 1], x))/(n^4 + 3*n^3 - 3*n^2 + 35*n + (2*n^3 + 6*n^2 - 20*n)*x + (n^2 + 5*n)*x^2)); 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-5939aa09e8ea2fc4","steps":["reverse","D+1"],"edges":["e-3594e486c04f1742","e-daa666daf8dffe6e"],"text":"A062190 \u2192 reverse \u2192 D+1"},"headline":"A062190 \u2192 reverse \u2192 D+1","chain":null}