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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":"5743196a67b42c27c283e919ab015330e21018d3818a656af32d0486f72f673c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-35273d9764d27bc4","role":"sequence","name":"basis_map_A013611 of Triangular array formed by the Mersenne 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:       8,        4;\n  2:      46,       64,       16;\n  3:     224,      560,      384,        64;\n  4:     996,     3712,     4512,      2048,       256;\n  5:    4192,    21040,    37888,     30208,     10240,      1024;\n  6:   17016,   108032,   261504,    307200,    181760,     49152,     4096;\n  7:   67328,   518528,  1588224,   2501120,   2170880,   1019904,   229376,    16384;\n  8:  261392,  2370560,  8819456,  17555456,  20372480,  13991936,  5447680,  1048576,   65536;\n  9: 1000064, 10447168, 45846528, 110835712, 162340864, 148721664, 84410368, 28049408, 4718592, 262144;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+3*n+2*n*x-2*n*x^2)*P(n,x) = (4*n^3+12*n^2+16*n+8+(4*n^3+12*n^2+14*n+6)*x-(4*n+4)*x^2-(4*n+4)*x^3)*P(n-1,x) + (2*n^2+2*n+1+(8*n^2+8*n+2)*x+(4*n^2+4*n-8)*x^2-12*x^3-4*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[((4*n^3 + 12*n^2 + 16*n + 8 + (4*n^3 + 12*n^2 + 14*n + 6)*x - (4*n + 4)*x^2 - (4*n + 4)*x^3)*P[n - 1, x] + (2*n^2 + 2*n + 1 + (8*n^2 + 8*n + 2)*x + (4*n^2 + 4*n - 8)*x^2 - 12*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 + 3*n + 2*n*x - 2*n*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] = ((4*n^3 + 12*n^2 + 16*n + 8 + (4*n^3 + 12*n^2 + 14*n + 6)*x - (4*n + 4)*x^2 - (4*n + 4)*x^3)*v[i - 1] + (2*n^2 + 2*n + 1 + (8*n^2 + 8*n + 2)*x + (4*n^2 + 4*n - 8)*x^2 - 12*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 + 3*n + 2*n*x - 2*n*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-9b5c7e97e5191da6","steps":["basis A013611","D+1"],"edges":["e-6b7a7c8afbb896b5","e-8b07136ba2dfdaeb"],"text":"A110441 \u2192 basis A013611 \u2192 D+1"},"headline":"A110441 \u2192 basis A013611 \u2192 D+1","chain":null}