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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":"2a5782323f38ec172c00783c43d55bf29be65ecdf79abb3b504a0034f31e211a","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-84c761192e0db3c0","role":"sequence","name":"basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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:     9,     40,      16;\n  1:    22,    148,     256,       64;\n  2:    52,    464,    1360,     1408,      256;\n  3:   120,   1328,    5408,     9792,     7168,      1024;\n  4:   272,   3584,   18560,    47616,    61696,     34816,      4096;\n  5:   608,   9280,   58112,   192000,   355840,    357376,    163840,     16384;\n  6:  1344,  23296,  170752,   688128,  1653760,   2387968,   1953792,    753664,     65536;\n  7:  2944,  57088,  478720,  2272256,  6694912,  12587008,  14852096,  10240000,   3407872,    262144;\n  8:  6400, 137216, 1294336,  7061504, 24600576,  56885248,  87621632,  87293952,  51970048,  15204352,  1048576;\n  9: 13824, 324608, 3399680, 20938752, 84099072, 230522880, 437649408, 570163200, 491126784, 257163264, 67108864, 4194304;\n","recurrence_sentence":"The row polynomials satisfy (2*n+7+(8*n+14)*x+8*x^2)*P(n,x) = (4*n+18+(24*n+80)*x+(32*n+104)*x^2+32*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 40*x + 16*x^2.","mathematica":"P[0, x_] := 9 + 40*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 18 + (24*n + 80)*x + (32*n + 104)*x^2 + 32*x^3)*P[n - 1, x])/(2*n + 7 + (8*n + 14)*x + 8*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 9 + 40*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 18 + (24*n + 80)*x + (32*n + 104)*x^2 + 32*x^3)*v[i - 1])/(2*n + 7 + (8*n + 14)*x + 8*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-7717563017cb9e5f","steps":["basis A013611","D+1"],"edges":["e-d3bf3843e93b142d","e-722aa21cdfa32da6"],"text":"A159854 \u2192 basis A013611 \u2192 D+1"},"headline":"A159854 \u2192 basis A013611 \u2192 D+1","chain":null}