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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":"6261be949bc2151fd52b8ae8e285a1f896aaf73a60c2d6a91355f484b31e6f67","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ba8de4e02acfb6ca","role":"sequence","name":"Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1;\n   2:   4,    3;\n   3:  11,   20,     5;\n   4:  22,   91,    56,      7;\n   5:  37,  288,   378,    120,      9;\n   6:  56,  715,  1716,   1122,    220,     11;\n   7:  79, 1508,  5863,   6864,   2717,    364,     13;\n   8: 106, 2835, 16380,  30745,  21450,   5733,    560,    15;\n   9: 137, 4896, 39508, 109616, 121550,  56576,  10948,   816,   17;\n  10: 172, 7923, 85272, 329460, 537472, 394706, 131784, 19380, 1140, 19;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-20*n^2+35*n-21+(8*n^2-34*n+33)*x+(4*n-6)*x^2)*P(n,x) = (4*n^3-12*n^2+13*n-6+(12*n^3-44*n^2+25*n+21)*x+(24*n^2-86*n+75)*x^2+(12*n-18)*x^3)*P(n-1,x) + ((8*n^2-16*n+2)*x-(8*n^2-32*n+30)*x^2-(16*n-36)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 12*n^2 + 13*n - 6 + (12*n^3 - 44*n^2 + 25*n + 21)*x + (24*n^2 - 86*n + 75)*x^2 + (12*n - 18)*x^3)*P[n - 1, x] + ((8*n^2 - 16*n + 2)*x - (8*n^2 - 32*n + 30)*x^2 - (16*n - 36)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 20*n^2 + 35*n - 21 + (8*n^2 - 34*n + 33)*x + (4*n - 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^3 - 12*n^2 + 13*n - 6 + (12*n^3 - 44*n^2 + 25*n + 21)*x + (24*n^2 - 86*n + 75)*x^2 + (12*n - 18)*x^3)*v[i - 1] + ((8*n^2 - 16*n + 2)*x - (8*n^2 - 32*n + 30)*x^2 - (16*n - 36)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 20*n^2 + 35*n - 21 + (8*n^2 - 34*n + 33)*x + (4*n - 6)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["D+1"],"edges":["e-c924eee3366ab2fb"],"text":"A091042 \u2192 D+1"},"headline":"A091042 \u2192 D+1","chain":null}