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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":"c9d775057a4465071ac376dcb4b8c8e566ac4852bb7284afda10057030bcdbc4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a18927332224e18f","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:     1;\n  2:     6,      4;\n  3:    25,     52,      16;\n  4:    89,    332,     320,      64;\n  5:   294,   1604,    2784,    1728,     256;\n  6:   929,   6696,   17136,   19072,    8704,    1024;\n  7:  2849,  25560,   87120,  142976,  116480,   41984,    4096;\n  8:  8550,  91844,  392352,  859200, 1030400,  660480,  196608,  16384;\n  9: 25241, 315932, 1625120, 4471232, 7153920, 6736896, 3555328, 901120, 65536;\n","recurrence_sentence":"The row polynomials satisfy (128*n^3-576*n^2+920*n-516+(64*n-96)*x-(128*n-192)*x^2)*P(n,x) = (192*n^3-448*n^2+308*n-96+(256*n^3-512*n^2+144*n+80)*x-(320*n-384)*x^2-(256*n-256)*x^3)*P(n-1,x) + (80*n^2-160*n+15+(384*n^2-768*n-48)*x+(256*n^2-512*n-608)*x^2-768*x^3-256*x^4)*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[((192*n^3 - 448*n^2 + 308*n - 96 + (256*n^3 - 512*n^2 + 144*n + 80)*x - (320*n - 384)*x^2 - (256*n - 256)*x^3)*P[n - 1, x] + (80*n^2 - 160*n + 15 + (384*n^2 - 768*n - 48)*x + (256*n^2 - 512*n - 608)*x^2 - 768*x^3 - 256*x^4)*D[P[n - 1, x], x])/(128*n^3 - 576*n^2 + 920*n - 516 + (64*n - 96)*x - (128*n - 192)*x^2)]];\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] = ((192*n^3 - 448*n^2 + 308*n - 96 + (256*n^3 - 512*n^2 + 144*n + 80)*x - (320*n - 384)*x^2 - (256*n - 256)*x^3)*v[i - 1] + (80*n^2 - 160*n + 15 + (384*n^2 - 768*n - 48)*x + (256*n^2 - 512*n - 608)*x^2 - 768*x^3 - 256*x^4)*deriv(v[i - 1], x))/(128*n^3 - 576*n^2 + 920*n - 516 + (64*n - 96)*x - (128*n - 192)*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-0eba401b117f16a9","steps":["basis A013611","D+1"],"edges":["e-39137b152517cefd","e-51cde39c907e778d"],"text":"A085478 \u2192 basis A013611 \u2192 D+1"},"headline":"A085478 \u2192 basis A013611 \u2192 D+1","chain":null}