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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":"ea0075bef10bd1aea4d234a554281c7adaa26d1fc15b33a5fd45c08017342b6b","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-efe494309672b404","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:     1;\n  2:    12,      2;\n  3:    77,     48,       3;\n  4:   336,    468,     120,       4;\n  5:  1089,   2904,    1650,     240,       5;\n  6:  2860,  12870,   13728,    4420,     420,      6;\n  7:  6461,  44616,   79365,   47320,    9975,    672,     7;\n  8: 13056, 129064,  350064,  344760,  133280,  19992,  1008,    8;\n  9: 24225, 325584, 1259700, 1881152, 1191870, 325584, 36708, 1440, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-28*n^3+81*n^2-102*n+45+(8*n^3-46*n^2+77*n-39)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (4*n^4-8*n^3+11*n^2+8*n+(12*n^4-64*n^3+143*n^2-175*n+48)*x+(24*n^3-114*n^2+161*n-44)*x^2+(12*n^2-26*n+8)*x^3)*P(n-1,x) + ((8*n^3-28*n^2+44*n-18)*x-(8*n^3-44*n^2+88*n-36)*x^2-(16*n^2-52*n+22)*x^3-(8*n-4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^4 - 8*n^3 + 11*n^2 + 8*n + (12*n^4 - 64*n^3 + 143*n^2 - 175*n + 48)*x + (24*n^3 - 114*n^2 + 161*n - 44)*x^2 + (12*n^2 - 26*n + 8)*x^3)*P[n - 1, x] + ((8*n^3 - 28*n^2 + 44*n - 18)*x - (8*n^3 - 44*n^2 + 88*n - 36)*x^2 - (16*n^2 - 52*n + 22)*x^3 - (8*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 28*n^3 + 81*n^2 - 102*n + 45 + (8*n^3 - 46*n^2 + 77*n - 39)*x + (4*n^2 - 10*n + 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^4 - 8*n^3 + 11*n^2 + 8*n + (12*n^4 - 64*n^3 + 143*n^2 - 175*n + 48)*x + (24*n^3 - 114*n^2 + 161*n - 44)*x^2 + (12*n^2 - 26*n + 8)*x^3)*v[i - 1] + ((8*n^3 - 28*n^2 + 44*n - 18)*x - (8*n^3 - 44*n^2 + 88*n - 36)*x^2 - (16*n^2 - 52*n + 22)*x^3 - (8*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 28*n^3 + 81*n^2 - 102*n + 45 + (8*n^3 - 46*n^2 + 77*n - 39)*x + (4*n^2 - 10*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["D","D+1"],"edges":["e-1287908e85a1078d","e-cebe6ba2eda79a13"],"text":"A103327 \u2192 D \u2192 D+1"},"headline":"A103327 \u2192 D \u2192 D+1","chain":null}