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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":"fbf8af648c6e9861c829b2048d40a1952f1a5538c3a1116a82a0179f8b02fc79","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-19327cf22807de6e","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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:             1035,               748,               121;\n  1:            29624,             34155,             12342,              1331;\n  2:           815189,           1303456,            751410,            181016,             14641;\n  3:         21827598,          44835395,          35845040,          13775850,           2488970,            161051;\n  4:        572834527,        1440621468,        1479568035,         788590880,         227301525,          32854404,           1771561;\n  5:      14803588900,       44108258579,       55463926518,       37975579565,       15180374440,        3500443485,         421631518,         19487171;\n  6:     377935624617,     1302715823200,     1940763377476,     1626941844528,      835462750430,      267174590144,       51339837780,       5300510512,        214358881;\n  7:    9553940204282,    37415626837083,    64484433248400,    64045191456708,    40266810652068,    16542162458514,     4408380737376,     726091991460,      65593817586,      2357947691;\n  8:  239553303974579,  1050933422471020,  2057859476039565,  2364429219108000,  1761242765059470,   885869834345496,   303272978406090,   69274554444480,    9983764882575,    801702214940,   25937424601;\n  9: 5965417614765456, 28985949780924059, 63581472059496710, 83000332200262455, 71523983878017000, 42622074914439174, 17865041659300836, 5242290055305270, 1047777635972760, 134226172310175, 9700596800774, 285311670611;\n","recurrence_sentence":"The row polynomials satisfy (11*n+34+11*x)*P(n,x) = (253*n+1035+(121*n+748)*x+121*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1035 + 748*x + 121*x^2.","mathematica":"P[0, x_] := 1035 + 748*x + 121*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((253*n + 1035 + (121*n + 748)*x + 121*x^2)*P[n - 1, x])/(11*n + 34 + 11*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1035 + 748*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((253*n + 1035 + (121*n + 748)*x + 121*x^2)*v[i - 1])/(11*n + 34 + 11*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-344e4bfa9933088a","steps":["basis A007318","D+1"],"edges":["e-347b6efdbfa5233f","e-6aa963d257e13159"],"text":"A038337 \u2192 basis A007318 \u2192 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