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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":"993762d8731fc82ca886ddbf6aa5cd61f97199f2e44cd94ef6bd64aa85150d8a","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-ef179a3c7df58b5a","role":"sequence","name":"reversal of Coefficient triangle of certain polynomials N(4; 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:     6,      1;\n  2:    27,     14,      1;\n  3:    98,    105,     24,      1;\n  4:   294,    560,    264,     36,      1;\n  5:   756,   2310,   1920,    540,     50,      1;\n  6:  1722,   7812,  10350,   5100,    975,     66,     1;\n  7:  3564,  22638,  44352,  34650,  11550,   1617,    84,    1;\n  8:  6831,  58080, 158928, 182952,  96250,  23408,  2520,  104,   1;\n  9: 12298, 135135, 494208, 792792, 612612, 234234, 43680, 3744, 126, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^4+2*n^3-2*n^2+24*n+(2*n^3+4*n^2-16*n)*x+(n^2+4*n)*x^2)*P(n,x) = (n^4+8*n^3+21*n^2+40*n+80+(3*n^4+4*n^3+14*n^2+24*n-80)*x+(6*n^3+4*n^2-18*n+28)*x^2+(3*n^2+6*n-4)*x^3)*P(n-1,x) + ((2*n^3+3*n^2+8*n+16)*x-(2*n^3-n^2+4*n+24)*x^2-(4*n^2+2*n-12)*x^3-(2*n+4)*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 + 8*n^3 + 21*n^2 + 40*n + 80 + (3*n^4 + 4*n^3 + 14*n^2 + 24*n - 80)*x + (6*n^3 + 4*n^2 - 18*n + 28)*x^2 + (3*n^2 + 6*n - 4)*x^3)*P[n - 1, x] + ((2*n^3 + 3*n^2 + 8*n + 16)*x - (2*n^3 - n^2 + 4*n + 24)*x^2 - (4*n^2 + 2*n - 12)*x^3 - (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 - 2*n^2 + 24*n + (2*n^3 + 4*n^2 - 16*n)*x + (n^2 + 4*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 + 8*n^3 + 21*n^2 + 40*n + 80 + (3*n^4 + 4*n^3 + 14*n^2 + 24*n - 80)*x + (6*n^3 + 4*n^2 - 18*n + 28)*x^2 + (3*n^2 + 6*n - 4)*x^3)*v[i - 1] + ((2*n^3 + 3*n^2 + 8*n + 16)*x - (2*n^3 - n^2 + 4*n + 24)*x^2 - (4*n^2 + 2*n - 12)*x^3 - (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^4 + 2*n^3 - 2*n^2 + 24*n + (2*n^3 + 4*n^2 - 16*n)*x + (n^2 + 4*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-9516b04042fe64c0","steps":["reverse","D+1"],"edges":["e-6d0521392d648070","e-a863198114f1f5d9"],"text":"A062264 \u2192 reverse \u2192 D+1"},"headline":"A062264 \u2192 reverse \u2192 D+1","chain":null}