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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":"c2d8903a92e49f28c0fcd26c203c115dc67b33f5cda59b76688081aa2ff6d285","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-6ffee7a877aa82b8","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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:             493,             276,              36;\n  1:           10115,            8874,            2484,             216;\n  2:          201433,          242760,          106488,           19872,            1296;\n  3:         3925487,         6042990,         3641400,         1064880,          149040,            7776;\n  4:        75252421,       141317532,       108773820,        43696800,         9583920,         1073088,          46656;\n  5:      1424116571,      3160601682,      2967668172,      1522833480,       458816400,        80504928,        7511616,         279936;\n  6:     26672013745,     68357595408,     75854440368,     47482690752,     18274001760,      4404637440,      644039424,       51508224,       1679616;\n  7:    495278778311,   1440288742230,   1845655076016,   1365379926624,    641016325152,    197359219008,    39641736960,     4968304128,     347680512,     10077696;\n  8:   9131266490269,  29716726698660,  43208662266900,  36913101520320,  20480698899360,   7692195901824,  1973592190080,   339786316800,   37262280960,   2317870080,    60466176;\n  9: 167327493737267, 602663588357754, 980651981055780, 950590569871800, 609066175085280, 270345225471552, 84614154920064, 18608154935040, 2803237113600, 273256727040, 15297942528, 362797056;\n","recurrence_sentence":"The row polynomials satisfy (6*n+23+6*x)*P(n,x) = (102*n+493+(36*n+276)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 493 + 276*x + 36*x^2.","mathematica":"P[0, x_] := 493 + 276*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((102*n + 493 + (36*n + 276)*x + 36*x^2)*P[n - 1, x])/(6*n + 23 + 6*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 493 + 276*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((102*n + 493 + (36*n + 276)*x + 36*x^2)*v[i - 1])/(6*n + 23 + 6*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-1f2dec04776afb2b","steps":["x \u21a6 x+1","D+1"],"edges":["e-653ba39db5056185","e-147853ae8a2be2a7"],"text":"A038320 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038320 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}