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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":"5deb032ddf957c2936c8c19955b192465cf0818e3fcf2867855a12891bbd3f14","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-b8c0cc94a00a4dd8","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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:         13,          4;\n  2:        153,        104,          16;\n  3:       1701,       1836,         624,          64;\n  4:      18225,      27216,       14688,        3328,        256;\n  5:     190269,     364500,      272160,       97920,      16640,       1024;\n  6:    1948617,    4566456,     4374000,     2177280,     587520,      79872,      4096;\n  7:   19663317,   54561276,    63930384,    40824000,   15240960,    3290112,    372736,    16384;\n  8:  196101729,  629226144,   872980416,   681924096,  326592000,   97542144,  17547264,  1703936,   65536;\n  9: 1937102445, 7059662244, 11326070592, 10475764992, 6137316864, 2351462400, 585252864, 90243072, 7667712, 262144;\n","recurrence_sentence":"The row polynomials satisfy (4*n+5+4*x)*P(n,x) = (36*n+81+(16*n+72)*x+16*x^2)*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[((36*n + 81 + (16*n + 72)*x + 16*x^2)*P[n - 1, x])/(4*n + 5 + 4*x)]];\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] = ((36*n + 81 + (16*n + 72)*x + 16*x^2)*v[i - 1])/(4*n + 5 + 4*x)); 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-366b2525683554ec","steps":["basis A013611","D+1"],"edges":["e-b30dba0e9f8b339c","e-ef47b11719ad9b10"],"text":"A038279 \u2192 basis A013611 \u2192 D+1"},"headline":"A038279 \u2192 basis A013611 \u2192 D+1","chain":null}