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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":"1ef16cf061a93a059972a6256039afe018da64c8390a0c55e636c55587cc8d57","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-9aa9e3dd2403bca2","role":"sequence","name":"shift_x_plus_one of Riordan array ((1+x)^3,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: 20,  24,    9,     1;\n  1: 28,  56,   39,    11,     1;\n  2: 36, 104,  113,    57,    13,     1;\n  3: 44, 168,  255,   195,    78,    15,     1;\n  4: 52, 248,  489,   513,   306,   102,    17,     1;\n  5: 60, 344,  839,  1131,   915,   450,   129,    19,    1;\n  6: 68, 456, 1329,  2201,  2271,  1503,   631,   159,   21,    1;\n  7: 76, 584, 1983,  3907,  4928,  4137,  2324,   853,  192,   23,   1;\n  8: 84, 728, 2825,  6465,  9668,  9884,  7014,  3430, 1120,  228,  25,  1;\n  9: 92, 888, 3879, 10123, 17541, 21204, 18270, 11250, 4878, 1436, 267, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n+3+(n+5)*x+x^2)*P(n,x) = (2*n+5+(3*n+11)*x+(n+7)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 20 + 24*x + 9*x^2 + x^3.","mathematica":"P[0, x_] := 20 + 24*x + 9*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 5 + (3*n + 11)*x + (n + 7)*x^2 + x^3)*P[n - 1, x])/(2*n + 3 + (n + 5)*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 20 + 24*x + 9*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 5 + (3*n + 11)*x + (n + 7)*x^2 + x^3)*v[i - 1])/(2*n + 3 + (n + 5)*x + 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-f48bed27dbc29c5f","steps":["P\u2099 + P\u2099\u208a\u2081","D+1"],"edges":["e-0331d381cbbd8a41","e-ad8c0cf33bbf2d38"],"text":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081 \u2192 D+1"},"headline":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081 \u2192 D+1","chain":null}