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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"cbfd395df419eae4968aab9d349594c7203ab6fc923a59b9dcac7119ddff0092","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-4b2df01c2184142e","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","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:             1332,              2576,                784;\n  1:            33048,            111888,             108192,              21952;\n  2:           758160,           3701376,            6265728,            4039168,              614656;\n  3:         16586208,         106142400,          259096320,          292400640,           141370880,            17210368;\n  4:        351459648,        2786482944,         8915961600,        14509393920,         12280826880,          4750061568,           481890304;\n  5:       7278615936,       68886091008,       273075328512,       582509491200,        710960302080,        481408413696,        155168677888,         13492928512;\n  6:     148157247744,     1630409969664,      7715242192896,     20389624528896,      32620531507200,      31851021533184,      17972580777984,       4965397692416,        377801998336;\n  7:    2975389355520,    37335626431488,    205431656177664,    648080344203264,    1284546345320448,    1644074787962880,    1337742904393728,     647012908007424,     156410027311104,     10578455953408;\n  8:   59111068529664,   833109019545600,   5226987700408320,  19173621243248640,   45365624094228480,   71934595337945088,   76723490104934400,   53509716175749120,   22645451780259840,   4866089738567680,    296196766695424;\n  9: 1163972315879424, 18206209107136512, 128298789010022400, 536637403908587520, 1476368835730145280, 2794522444204474368, 3692642560681181184, 3375833564617113600, 2060124072766341120, 774977683146670080, 149875563947884544, 8293509467471872;\n","recurrence_sentence":"The row polynomials satisfy (14*n+23+14*x)*P(n,x) = (252*n+666+(392*n+1288)*x+392*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1332 + 2576*x + 784*x^2.","mathematica":"P[0, x_] := 1332 + 2576*x + 784*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((252*n + 666 + (392*n + 1288)*x + 392*x^2)*P[n - 1, x])/(14*n + 23 + 14*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1332 + 2576*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((252*n + 666 + (392*n + 1288)*x + 392*x^2)*v[i - 1])/(14*n + 23 + 14*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-7dcf11eddf2b9a8a","steps":["basis 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