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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":"d80f8ae19a42efda0ee71157583429cdc3e8655be16cd32135e0e65f02ee06a1","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-2fe6a26f73f75da6","role":"sequence","name":"basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:    21,     20,       4;\n  1:    60,    106,      56,       8;\n  2:   156,    416,     388,     144,      16;\n  3:   384,   1384,    1904,    1224,     352,       32;\n  4:   912,   4160,    7648,    7168,    3536,      832,      64;\n  5:  2112,  11680,   27008,   33600,   24000,     9632,    1920,     128;\n  6:  4800,  31232,   87232,  135936,  128320,    74240,   25152,    4352,     256;\n  7: 10752,  80512,  263936,  495488,  584192,   445312,  216832,   63616,    9728,    512;\n  8: 23808, 201728,  759808, 1671168, 2369024,  2250752, 1440768,  606208,  156928,  21504,  1024;\n  9: 52224, 494080, 2103296, 5308416, 8804352, 10042368, 7999488, 4417536, 1637376, 379392, 47104, 2048;\n","recurrence_sentence":"The row polynomials satisfy (3*n+4+(2*n+7)*x+2*x^2)*P(n,x) = (6*n+14+(10*n+32)*x+(4*n+22)*x^2+4*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 21 + 20*x + 4*x^2.","mathematica":"P[0, x_] := 21 + 20*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 14 + (10*n + 32)*x + (4*n + 22)*x^2 + 4*x^3)*P[n - 1, x])/(3*n + 4 + (2*n + 7)*x + 2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 21 + 20*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 14 + (10*n + 32)*x + (4*n + 22)*x^2 + 4*x^3)*v[i - 1])/(3*n + 4 + (2*n + 7)*x + 2*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":["basis A013609","D+1"],"edges":["e-8866dbf598c953fc","e-c8fdfc6f6c264f1b"],"text":"A391533 \u2192 basis A013609 \u2192 D+1"},"headline":"A391533 \u2192 basis A013609 \u2192 D+1","chain":null}