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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":"752d109a99ad467d7253469399e9a02293f8138d80f5ec017a562f8fb6ce38c8","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-20bccc4c38847de1","role":"sequence","name":"basis_map_A013609 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 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:          1;\n  1:          3,          2;\n  2:         14,         18,          4;\n  3:        106,        174,         72,          8;\n  4:       1116,       2108,       1164,        240,        16;\n  5:      15068,      31308,      20620,       5800,       720,        32;\n  6:     248368,     553920,     412416,     141160,     24240,      2016,       64;\n  7:    4835472,   11401472,    9307536,    3660552,    775040,     90048,     5376,    128;\n  8:  108593504,  267961152,  235241296,  102794720,  25290384,   3657472,   307328,  13824,   256;\n  9: 2763526240, 7084714176, 6600666672, 3135940800, 865348848, 146633760, 15447936, 983808, 34560, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(12*n-20)*x+4*x^2)*P(n-2,x) + (6*n-7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*P[n - 2, x] + (6*n - 7 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*v[i - 2] + (6*n - 7 + 4*x)*v[i - 1]); 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-1a0f876fa1441737","steps":["basis A013609","D+1"],"edges":["e-d3f9f2e05c32db01","e-f399aeb62fe3cbee"],"text":"basis-map coefficients of A371080, shifted \u2192 basis A013609 \u2192 D+1"},"headline":"basis-map coefficients of A371080, shifted \u2192 basis A013609 \u2192 D+1","chain":null}