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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":"4954dfad32c3732547b8199bcfcb920ab83159769d833c0e250c237ac3136557","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-97afe0d58b0705a8","role":"sequence","name":"derivative of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:     0;\n  2:     6;\n  3:    56,     24;\n  4:   252,    360,      60;\n  5:   792,   2376,    1320,     120;\n  6:  2002,  10296,   12012,    3640,     210;\n  7:  4368,  34320,   68640,   43680,    8400,    336;\n  8:  8568,  95472,  291720,  318240,  128520,  17136,   504;\n  9: 15504, 232560, 1007760, 1679600, 1162800, 325584, 31920, 720;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (2*n^2+3*n+1+(6*n^2-15*n+3)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n + 1 + (6*n^2 - 15*n + 3)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 + 3*n + 1 + (6*n^2 - 15*n + 3)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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-2245b35ec7cd4857","steps":["D","D"],"edges":["e-2a66c334394f3697","e-cd5c78ee50ecc887"],"text":"A091044 \u2192 D \u2192 D"},"headline":"A091044 \u2192 D \u2192 D","chain":null}