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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":"75fdaad22f1ac10fe85e646062ce30b3f7f038988de405ed45a3f73a002cc3d7","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:  1;\n  1:  4,   3;\n  2:  7,  22,    8;\n  3: 10,  59,   88,    21;\n  4: 13, 114,  321,   310,     55;\n  5: 16, 187,  788,  1434,   1020,    144;\n  6: 19, 278, 1570,  4284,   5721,   3216,    377;\n  7: 22, 387, 2748, 10075,  20170,  21207,   9848,    987;\n  8: 25, 514, 4403, 20346,  54780,  86250,  74657,  29522,  2584;\n  9: 28, 659, 6616, 36960, 125580, 265929, 344212, 252958, 87080, 6765;\n","recurrence_sentence":"The row polynomials satisfy (12*n^3-14*n^2+4*n+(10*n^3-6*n^2-8*n)*x+(20*n^2-28*n)*x^2+10*n*x^3)*P(n,x) = (12*n^3-2*n^2-2*n+(64*n^3-36*n^2-38*n)*x+(75*n^3-27*n^2-111*n+19)*x^2+(25*n^3+75*n^2-121*n+37)*x^3+(50*n^2-25*n+5)*x^4+(25*n-25)*x^5)*P(n-1,x) - ((6*n^2+2*n)*x+(41*n^2+19*n-2)*x^2+(60*n^2+62*n-11)*x^3+(25*n^2+95*n+1)*x^4+(50*n+35)*x^5+25*x^6)*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[((12*n^3 - 2*n^2 - 2*n + (64*n^3 - 36*n^2 - 38*n)*x + (75*n^3 - 27*n^2 - 111*n + 19)*x^2 + (25*n^3 + 75*n^2 - 121*n + 37)*x^3 + (50*n^2 - 25*n + 5)*x^4 + (25*n - 25)*x^5)*P[n - 1, x] - ((6*n^2 + 2*n)*x + (41*n^2 + 19*n - 2)*x^2 + (60*n^2 + 62*n - 11)*x^3 + (25*n^2 + 95*n + 1)*x^4 + (50*n + 35)*x^5 + 25*x^6)*D[P[n - 1, x], x])/(12*n^3 - 14*n^2 + 4*n + (10*n^3 - 6*n^2 - 8*n)*x + (20*n^2 - 28*n)*x^2 + 10*n*x^3)]];\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] = ((12*n^3 - 2*n^2 - 2*n + (64*n^3 - 36*n^2 - 38*n)*x + (75*n^3 - 27*n^2 - 111*n + 19)*x^2 + (25*n^3 + 75*n^2 - 121*n + 37)*x^3 + (50*n^2 - 25*n + 5)*x^4 + (25*n - 25)*x^5)*v[i - 1] - ((6*n^2 + 2*n)*x + (41*n^2 + 19*n - 2)*x^2 + (60*n^2 + 62*n - 11)*x^3 + (25*n^2 + 95*n + 1)*x^4 + (50*n + 35)*x^5 + 25*x^6)*deriv(v[i - 1], x))/(12*n^3 - 14*n^2 + 4*n + (10*n^3 - 6*n^2 - 8*n)*x + (20*n^2 - 28*n)*x^2 + 10*n*x^3)); 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+1"],"edges":["e-a162795980dd8c54","e-80651527fe670bbe"],"text":"A125662 \u2192 reverse \u2192 D+1"},"headline":"A125662 \u2192 reverse \u2192 D+1","chain":null}