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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":"a7dd16e241ad722a492a36e1a8c9d7a653960fbe5be5fbf97284a2470582bfab","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-7ba399ade8b53e3c","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, s_3(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:           7,            4;\n  2:          54,           68,           16;\n  3:         558,          972,          480,           64;\n  4:        7344,        15480,        10800,         2944,         256;\n  5:      117936,       283176,       244080,        95040,       16640,        1024;\n  6:     2239488,      5924016,      5914944,      2882880,      725760,       89088,       4096;\n  7:    49128768,    140220720,    156328704,     89280576,    28385280,     5031936,     458752,     16384;\n  8:  1223180352,   3714237504,   4517000640,   2908410624,  1092559104,   246177792,   32514048,   2293760,    65536;\n  9: 34084027584, 109011697344, 142327160640, 100780692480, 42994748160, 11518682112, 1943101440, 199065600, 11206656, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(24*n-24)*x+16*x^2)*P(n-2,x) + (6*n-3+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (24*n - 24)*x + 16*x^2)*P[n - 2, x] + (6*n - 3 + 8*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] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (24*n - 24)*x + 16*x^2)*v[i - 2] + (6*n - 3 + 8*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-8229dbdd6a4b5cd9","steps":["basis A013611","D+1"],"edges":["e-6ce7f219250b8b1f","e-912241a40d4e9023"],"text":"A225470 \u2192 basis A013611 \u2192 D+1"},"headline":"A225470 \u2192 basis A013611 \u2192 D+1","chain":null}