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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":"06f82c4adc5e2e6a323a9142972f2a28809676706550666973c69ff9284549b2","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-1c1963609d82ab65","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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,           1;\n  2:          20,          10,           1;\n  3:         212,         128,          21,          1;\n  4:        3088,        2096,         440,         36,         1;\n  5:       57264,       42224,       10360,       1120,        55,        1;\n  6:     1290048,     1013216,      277744,      35920,      2380,       78,      1;\n  7:    34206528,    28258432,     8428784,    1239504,     99820,     4480,    105,     1;\n  8:  1043493120,   898868736,   286796544,   46531968,   4314464,   238112,   7728,   136,   1;\n  9: 35997684480, 32129604864, 10837098240, 1902099968, 196552608, 12607392, 507696, 12480, 171, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-48*n+36+(8*n-12)*x+x^2)*P(n-2,x) + (8*n-8+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 48*n + 36 + (8*n - 12)*x + x^2)*P[n - 2, x] + (8*n - 8 + 2*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 + x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 48*n + 36 + (8*n - 12)*x + x^2)*v[i - 2] + (8*n - 8 + 2*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-4172472abe953cea","steps":["x \u21a6 x+1","D+1"],"edges":["e-2a810a02045f5dfd","e-cfad9b5a4f34d686"],"text":"A290319 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A290319 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}