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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":"e77a1e08493711c799f764f312f9862061ee06e03729f0f169161d55af5dfa11","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c4e1c7f7d91f2e30","role":"sequence","name":"reversal of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:  3;\n  2:  6,  16;\n  3:  9,  50,   63;\n  4: 12, 102,  270,   220;\n  5: 15, 172,  702,  1200,    720;\n  6: 18, 260, 1440,  3804,   4770,   2262;\n  7: 21, 366, 2565,  9220,  17865,  17634,   6909;\n  8: 24, 490, 4158, 18960,  50040,  76242,  61950,  20672;\n  9: 27, 632, 6300, 34860, 116865, 242556, 303786, 209560, 60885;\n","recurrence_sentence":"The row polynomials satisfy (12*n^2-18*n+6+(10*n^2-30*n+20)*x)*P(n,x) = (12*n^2-6*n+(64*n^2-78*n-4)*x+(75*n^2-165*n+30)*x^2+(25*n^2-75*n+50)*x^3)*P(n-1,x) - (6*n*x+(41*n-5)*x^2+(60*n-30)*x^3+(25*n-25)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 6*n + (64*n^2 - 78*n - 4)*x + (75*n^2 - 165*n + 30)*x^2 + (25*n^2 - 75*n + 50)*x^3)*P[n - 1, x] - (6*n*x + (41*n - 5)*x^2 + (60*n - 30)*x^3 + (25*n - 25)*x^4)*D[P[n - 1, x], x])/(12*n^2 - 18*n + 6 + (10*n^2 - 30*n + 20)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^2 - 6*n + (64*n^2 - 78*n - 4)*x + (75*n^2 - 165*n + 30)*x^2 + (25*n^2 - 75*n + 50)*x^3)*v[i - 1] - (6*n*x + (41*n - 5)*x^2 + (60*n - 30)*x^3 + (25*n - 25)*x^4)*deriv(v[i - 1], x))/(12*n^2 - 18*n + 6 + (10*n^2 - 30*n + 20)*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-f287bed89e07609f","steps":["reverse","D"],"edges":["e-a162795980dd8c54","e-6ef1ad404f22492e"],"text":"A125662 \u2192 reverse \u2192 D"},"headline":"A125662 \u2192 reverse \u2192 D","chain":null}