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for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"79cdf4d2bda189c217e130db5938101f2a64a5cada2b9d332bf1836a18dd9e96","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f4c6c763b9527d17","role":"sequence","name":"basis_map_A008277 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:            5,            1;\n  2:           46,           15,           1;\n  3:          613,          258,          30,          1;\n  4:        10679,         5317,         831,         50,         1;\n  5:       229576,       129315,       24580,       2035,        75,        1;\n  6:      5868715,      3640926,      799080,      81860,      4215,      105,       1;\n  7:    173833661,    116694655,    28657986,    3435460,    221480,     7791,     140,     1;\n  8:   5853205468,   4198767903,  1130030545,  153220466,  11669175,   518378,   13258,   180,   1;\n  9: 220767370219, 167666446872, 48729620556, 7305525885, 634248531, 33520599, 1089732, 21186, 225, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+1+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n + 1 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], 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] = (4*n + 1 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], 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-9c5c208df8605708","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-3a6a7b6037078bc1","e-ad9851a5229f0685"],"text":"A225471 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A225471 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}